How do Reformed Protestants interpret Matthew 18:18?

The Presbyterian and Congregational view is different on this matter. In the 1689 London Baptist Confession of Faith, used by Congregational Reformed Churches, it says in Chapter 26 (Of the Church), paragraph 7:

To each of these churches thus gathered, according to his mind declared in his word, he hath given all that power and authority, which is in any way needful for their carrying on that order in worship and discipline, which he hath instituted for them to observe; with commands and rules for the due and right exerting, and executing of that power.

To each of these churches…. he hath given all that power and authority… for their carrying on that order in worship and discipline, referring to Matthew 18: 17-20. All authority for discipline has been given to the entire congregation, the local church.

While the Presbyterians in the Westminster confession chapter 30 (Of Church Censures), paragraph 1 and 2 say:

1. The Lord Jesus, as king and head of his Church, hath therein appointed a government in the hand of Church officers, distinct from the civil magistrate.
2. To these officers the keys of the Kingdom of Heaven are committed, by virtue whereof they have power respectively to retain and remit sins, to shut that kingdom against the impenitent, both by the word and censures; and to open it unto penitent sinners, by the ministry of the gospel, and by absolution from censures, as occasion shall require.

The keys to the kingdom, responsibility and authority have been given not to the congregation but to Church officers. In chapter 31 they raise the need for an external authority above the local church, to exercise its directives, government and order of worship.

But what does this have to do with Matthew 18:18?

All this has to do with the discipline of the Church, the local church (in the case of the Congregational) or the Church officers (Presbyterians) have the capacity and the same authority given to Peter in Matthew 16:19 to excommunicate a person who persists in his sin, judging him according to Scripture and having him as if she were a heathen man and a publican, as an unbeliever. It is the members or the officers who can decide whether the person is a member of the Church of Christ or not, after having passed the entire discipline system. They are the ones called to recognize or refuse to recognize people as belonging to the kingdom.

The apostle Paul speaks to the congregation of the Corinthian Church and holds it responsible for tolerating the evil that was generated within it (1 Cor. 5:1-5). Indicating that where the presence of Christ is promised to be in believers when they make decisions of this type. The removal of the person who has not repented of his sin, places him in the kingdom of Satan, but that does not imply eternal condemnation. At the end of the chapter he indicates that it is the duty of the church to judge its members.

For what have I to do with judging outsiders? Is it not those inside the church whom you are to judge? God judges those outside. Purge the evil person from among you. [1 Cor. 5:12-13]

St. Augustin on the Sermon XXXII explains it this way.

“take with thee one or two more, that in the mouth of two or three witnesses every word may be established; and if he will not hear them, refer it to the Church; but if he will not hear the Church, let him be unto thee as an heathen man and a publican.” [Matt. 18:16-17] Reckon him no more amongst the number of thy brethren. But yet neither is his salvation on that account to be neglected. For the very heathen, that is, the Gentiles and Pagans, we do not reckon among the number of brethren; but yet are we ever seeking their salvation. This then have we heard the Lord so advising, and with such great carefulness enjoining, that He even added this immediately, “Verily I say unto you, Whatsoever ye shall bind on earth, shall be bound in heaven; and whatsoever ye shall loose on earth, shall be loosed in heaven.” [Matt. 18:18] Thou hast begun to hold thy brother for a publican; “thou bindest him on earth;” but see that thou bind him justly. For unjust bonds justice doth burst asunder. But when thou hast corrected, and been “reconciled to thy brother,” thou hast “loosed him on earth.” And when “thou shalt have loosed him on earth, he shall be loosed in heaven also.” Thus thou doest a great thing, not for thyself, but for him; for a great injury had he done, not to thee, but to himself.

Calvin writes a commentary which address this passage in some detail. He summarises it as follows:

The substance of it is this: Whoever, after committing a crime, humbly confesses his fault, and entreats the Church to forgive him, is absolved not only by men, but by God himself; and, on the other hand, whoever treats with ridicule the reproofs and threatenings of the Church, if he is condemned by her, the decision which men have given will be ratified in heaven.

Calvin's point is that since Christ is the head of the church, the church does Christ's will. If the church chooses to forgive or reject, that is because Christ has chosen to forgive or reject.

For, though Christ alone is the Judge of the world, yet he chooses to have ministers to proclaim his word. Besides, he wishes that his own decision should be pronounced by the Church; and thus he takes nothing from his own authority by employing the ministry of men, but it is Himself alone that looses and binds.

Calvin continues to discuss several objections to and particulars of this teaching, which I won't repeat here.

Here is the full text from Calvin's Commentary. I do not think any Presbyterian or Reformed would disagree.

What things soever you shall bind. He now repeats the same words which he had formerly used, (Matthew 16:19,) but in a different sense; for there he intended to maintain their authority in doctrine, but here he appoints discipline, which is an appendage to doctrine. There Christ declared that the preaching of the Gospel would not be without effect, but that the odor of it would either be life-giving or deadly, (2 Corinthians 2:15 :) here he affirms that, though wicked men ridicule the judgment of the Church, it will not be ineffectual. We must attend to this distinction, that there our Lord’s discourse relates to the preached word, but here to public censures and discipline. Let the reader go to that passage for the import of the metaphor, binding and loosin g. (561)

The substance of it is this: Whoever, after committing a crime, humbly confesses his fault, and entreats the Church to forgive him, is absolved not only by men, but by God himself; and, on the other hand, whoever treats with ridicule the reproofs and threatenings of the Church, if he is condemned by her, the decision which men have given will be ratified in heaven. If it be objected, that in this way God is made a sort of petty judge, who concurs in the sentence of mortal men, the reply is at hand. For when Christ maintains the authority of his Church, he does not diminish his own power or that of his Father, but, on the contrary, supports the majesty of his word. As in the former case (Matthew 16:19) he did not intend to confirm indiscriminately every kind of doctrine, but only that which had proceeded out of his mouth, so neither does he say in this place that every kind of decision will be approved and ratified, but only that in which he presides, and that too not only by his Spirit, but by his word. Hence it follows, that men do no injury to the authority of God, when they pronounce nothing but what comes from his mouth, and only endeavor faithfully to execute what he has commanded. For, though Christ alone is the Judge of the world, yet he chooses to have ministers to proclaim his word. (562) Besides, he wishes that his own decision should be pronounced by the Church; and thus he takes nothing from his own authority by employing the ministry of men, but it is Himself alone that looses and binds

But here a question arises. Since the Church endures many hypocrites, and likewise absolves (or looses) many whose professions of repentance are hypocritical, does it follow that such persons will be absolved (or loosed) in heaven? I reply, the discourse is addressed to those only who are truly and sincerely reconciled to the Church. For Christ, wishing to administer comfort to trembling consciences, and to relieve them from fear, declares that any who may have offended are freed from guilt in the sight of God, provided that they be reconciled to the Church For he has appointed this as the pledge of heavenly grace, which has no reference to hypocrites, who pervert the proper use of reconciliation, but awakens in the godly no ordinary confidence, when they hear that their sins are blotted out before God and angels, as soon as they have obtained forgiveness from the Church

In the other clause, Christ’s meaning is not at all ambiguous; for, since obstinate and haughty men are strongly inclined to despise the decision of the Church on this pretense, that they refuse to be subject to men — as wicked profligates often make bold appeals to the heavenly tribunal (563) — Christ, in order to subdue this obstinacy by terror, threatens that the condemnation, which is now despised by them, will be ratified in heaven. He encourages his followers, at the same time, to maintain proper severity, and not to yield to the wicked obstinacy of those who reject or shake off discipline. (564)

Hence, too, we may see how absurdly the Papists torture this passage to cloak every species of tyranny. That the right of excommunication is granted to the Church is certain, and is acknowledged by every person of sound judgment; but does it follow that any individual, even though not called by the Church, but elected (565) by a mitered and disguised beast, shall at his own caprice throw out the useless squibs of excommunications? (566) On the contrary, it is evident that the lawful government of the Church is committed to elders, and not only to the ministers of the word, but to those also who, taken from among the people, have been added to them for the superintendence of morals. And yet, not satisfied with this impudence, they endeavor even to prove from this passage that we must bear all the burdens which they shall impose. I do not mention that the power which has been granted to the Church is basely seized and carried off by those outrageous enemies of the Church; and I only mention that, since Christ speaks only about correcting offenders, those who by their laws ensnare souls are chargeable with not less folly than wickedness in abusing this passage. Of the same stamp is their defense of their auricular confession on this pretense; for if Christ intended that those who by their own fault had been brought even to a public sentence should be reconciled to the Church, he does not therefore lay an obligations (567) on every individual to pour his sins into the ear of the priest. But their fooleries are so ridiculous, that it is unnecessary to spend any longer time in refuting them.