It is to be noted what kind of elders are to be especially honoured and rewarded. It is those who toil in preaching and teaching. The elder whose service consisted only in words and discussion and argument is not in question here. Those whom the Church really honoured were the ones who worked to edify and build it up by preaching the truth and by educating the young and the new converts in the Christian way.

(2) It was Jewish law that no one should be condemned on the evidence of a single witness: ‘A single witness shall not suffice to convict a person of any crime or wrongdoing in connection with any offence that may be committed. Only on the evidence of two or three witnesses shall a charge be sustained’ (Deuteronomy 19:15). The Mishnah, the codified Rabbinic law, in describing the process of trial, says: ‘The second witness was likewise brought in and examined. If the testimony of the two was found to agree, the case for the defence was opened.’ If a charge was supported by the evidence of only one witness, it was held that there was no case to answer.

Quoted verse:
Deuteronomy 19:15 but I will read verses 15 through 21.  These scriptures are linked clearly to the instructions in Matthew 18 or the "offending brother resolution instructions."  The God-given concepts are equal.

Deuteronomy 19:15-21
15 One witness shall not rise up against a man for any iniquity, or for any sin, in any sin that he sinneth: at the mouth of two witnesses, or at the mouth of three witnesses, shall the matter be established.
16 If a false witness rise up against any man to testify against him that which is wrong;
17 Then both the men, between whom the controversy is, shall stand before the LORD, before the priests and the judges, which shall be in those days;
18 And the judges shall make diligent inquisition: and, behold, if the witness be a false witness, and hath testified falsely against his brother;
19 Then shall ye do unto him, as he had thought to have done unto his brother: so shalt thou put the evil away from among you.
20 And those which remain shall hear, and fear, and shall henceforth commit no more any such evil among you.
21 And thine eye shall not pity; but life shall go for life, eye for eye, tooth for tooth, hand for hand, foot for foot.

In later times, church regulations laid it down that the two witnesses must be Christian [Spirit-holding firstfruits], for it would have been easy for a malicious non-Christian to make up a false charge against a Christian elder in order to discredit him, and through him to discredit the Church. In the early days, the Church authorities did not hesitate to apply discipline; and Theodore of Mopseuestia, one of the early fathers who lived in the late fourth and early fifth centuries, points out how necessary this regulation was, because the elders were always liable to be disliked and were especially open to malicious attack ‘due to the retaliation by some who had been rebuked by them for sin’. Those who had been disciplined might well seek to get their own back by maliciously charging an elder with some irregularity or some sin.

The fact remains that this would be a happier world – and the Church, too, would be happier – if people would realize that it is nothing less than sin to spread stories of whose truth they are not sure. Irresponsible, slanderous and malicious talk does infinite damage and causes infinite heartbreak, and such talk will not go unpunished by God.

(3) Those who persist in sin are to be publicly rebuked [meaning within the congregation of firstfruits]. That public rebuke had a double value. It sobered sinners into a consideration of their ways, and it made others take care that they did not involve themselves in a similar humiliation. The threat of publicity is no bad thing if it keeps people on the right path, even through fear. A wise leader will know the time to keep things quiet and the time for public rebuke. But, whatever happens, the Church must never give the impression that it is condoning sin.

(4) Timothy is urged to administer his office without favoritism or prejudice. The biblical scholar B. S. Easton writes: ‘The well-being of every community depends on impartial discipline.’ Nothing does more harm than when some people are treated as if they could do no wrong and others as if they could do no right. Justice is a universal virtue, and the Church must surely never fall below the impartial standards which even the world demands.

(5) Timothy is warned not to be too hasty ‘in laying hands on any man’. That may mean one of two things.

(a) It may mean that he is not to be too quick in laying hands on any man to ordain him to office in the Church. Before people gain promotion in business, or in teaching, or in the army or the navy or the air force, they must prove that they deserve it. No one should ever start at the top. This is doubly important in the Church, for those who are raised to high office and then fail in it bring dishonour, not only on themselves, but also on the Church. In a critical world, the Church cannot be too careful in regard to the kind of men and women whom it chooses as its leaders.

(b) In the early Church, it was the custom to lay hands on a sinner who repented, who had given proof of repentance and who had returned to the fold of the Church. It is laid down: ‘As each sinner repents, and shows the fruits of repentance, lay hands on him, while all pray for him.’ The early Church historian Eusebius tells us that it was the ancient custom that repentant sinners should be received back with the laying on of hands and with prayer. If that is the meaning here, it will be a warning to Timothy not to be too quick to receive back anyone who has brought disgrace on the Church, to wait until the individual has shown genuine [repentance and invoking of Godly principles-fruit] and a true determination to live according to that declaration of repentance. That is not for a moment to say that such a person is to be held at arm’s length and treated with suspicion, but rather to be treated with all sympathy and with all help and guidance in the period of probation. But it is to say that membership of the Church is never to be treated lightly, and that people must show their [repentance] for the past and their determination for the future before they are received not into the fellowship of the Church but into its membership. The fellowship of the Church exists to help such people redeem themselves, but its membership is for those who have truly pledged their lives to Christ. ~Barclay Commentary

Now to the commentaries.

This verse is in three phrases:

1] Them that sin.
2] Rebuke before all.
3] That other also may fear.

Some commentaries combine the first two phrases.

1] Them that sin.

Them that sin — whether presbyters or laymen. ~Jamieson, Fausset, Brown

Them that sin — Referring to Elders, who, by reason of their public position (προεστῶτες), should receive public rebuke. ~Vincent's Word Studies

Them that sin — Elders. That sin - Scandalously, and are duly convicted. Rebuke before all - The church. ~John Wesley Explanatory Notes

Them that sin - The elders who continue to sin. ~Robertson's Word Pictures

Them that sin - Whether they be elders, or private members, if convicted, rebuke them publicly, for the sake of the lesson to others. Do not hush the matter up privately. ~People's New Testament

Them that sin - That have been proved to have committed sin - referring probably to the elders mentioned in the previous verse, but giving the direction so general a form that it might be applicable to others. ~Barnes Notes

Them that sin rebuke before all - That is, before the members of the Church; which was the custom of the Jews in their synagogues. But, if the words refer to the elders alone, then the transgressing elder is to be reproved before his fellows, and be tried by them. ~Adam Clarke

Them that sin rebuke before all - This the apostle adds to the above rule, to show that he was far from screening wicked ministers, or elders, guilty of flagitious crimes, and gross enormities: for these words, though they may be applied unto, and may hold good of all offenders, that are members of churches; yet they seem chiefly to regard elders, even such who sin, who continue to sin, who live in sin, in some notorious sin or another; which is evident and known, to the great scandal of religion, and dishonour of the Gospel: and so some read the words, "them that sin before all, rebuke"; not only admonish once and again, but degrade them from their office, and withdraw from them, as from other disorderly persons, and cut them off, and cast them out of the church, and that in a public manner; and so the Arabic version renders it, "before the congregation": which was done only in case of notorious offences: and which rule is observed by the Jews. ~John Gill

2] Rebuke before all.

Rebuke before all — publicly before the Church. Not until this “rebuke” was disregarded was the offender to be excommunicated. ~Jamieson, Fausset, Brown

Rebuke before all - Before all the church or congregation. The word “rebuke” properly denotes to reprove or reprehend. It means here that there should be a public statement of the nature of the offence, and such a censure as the case demanded. It extends only to spiritual censures. There is no power given of inflicting any punishment by fine or imprisonment. The power of the church, in such cases, is only to express its strong and decided disapprobation [dis-ap-ruh-bey-shuhn] of the wrong done, and, if the case demands it, of disowning the offending member or minister. This direction to “rebuke an offender before all,” may be easily reconciled with the direction in 1 Timothy 5:1, “Rebuke not an elder.” The latter refers to the private and pastoral conversation with an elder, and to the method in which he should be treated in such contact - to wit, with the feelings due to a father; the direction here refers to the manner in which an offender should be treated who has been proved to be guilty, and where the case has become public. Then there is to be a public expression of disapprobation. ~Barnes Notes

Quoted verse:
1 Timothy 5:1
The elders which are among you I exhort, who am also an elder, and a witness of the sufferings of Christ, and also a partaker of the glory that shall be revealed:

Those that sin rebuke before all - that is, thou needest not be so tender of other people, but rebuke them publicly.” Or “those that sin before all rebuke before all, that the plaster may be as wide as the wound, and that those who are in danger of sinning by the example of their fall may take warning by the rebuke given them for it, that others also may fear.” Observe, (1.) Public scandalous sinners must be rebuked publicly: as their sin has been public, and committed before many, or at least come to the hearing of all, so their reproof must be public, and before all. (2.) Public rebuke is designed for the good of others, that they may fear, as well as for the good of the party rebuked; hence it was ordered under the law that public offenders should receive public punishment, that all Israel might hear, and fear, and do no more wickedly. ~Matthew Henry

3] That other also may fear.

That others also may fear - that other members of the Church may have a wholesome fear of offending ~Jamieson, Fausset, Brown

That others also may fear - “may keep on having fear” (of exposure). Possibly, “the rest of the elders.” ~Robertson's Word Pictures

That others also may fear - That they may be kept from committing the same offence; compare 1 Peter 2:14. The end of punishment is not the gratification of the private feelings of him who administers it, but the prevention of crime. ~Barnes Notes

Quoted verse:
1 Peter 2:14
Or unto governors, as unto them that are sent by him for the punishment of evildoers, and for the praise of them that do well.

That others also may fear - This is the grand object of Church censures, to reclaim the transgressors, and to give warning to others. ~Adam Clarke

That others also may fear - that other elders, or other members of the church, or both, may fear to do the same evil things, lest they incur the same censure and punishment: the Syriac version reads, "other men"; and the Arabic version, "the rest of the people". The phrase seems to be taken out of Deuteronomy 13:11. ~John Gill

Quoted verse:
Deuteronomy 13:11
And all Israel shall hear, and fear, and shall do no more any such wickedness as this is among you.