Rev. Brian M. Abshire has written an excellent piece “On the Application of Matthew 18 to Public Sin,” which refutes the commonly-held view that sin must always be addressed privately.
As he demonstrates, where there is the sin of distorting gospel, and hence the well-being of the church is at stake, dealing with the matter–at least 0ften–is a public, not a private, matter. Here is an excerpt:
The common thread to all these “exceptions” to the first step of Matthew 18 is that there is more than the individual sinner’s welfare is at stake. This actually makes sense in the context of Matthew 18:15. The Majority Text of verse 15 adds the words “against you.”
Our Lord’s words should perhaps be best understood in reference to personal offences, rather than public sins. In the three “exceptions” we have been looking at, never is it a personal offence, but a public sin that affects the spiritual health and well being of the broader church.
In the 3 John situation, the integrity of the Apostolic message was being damaged and people were being intimidated or thrown out of the church by an ecclesiastical tyrant. In Galatians, Peter was in danger of splitting the Antioch church apart because of his own bad example.
Hymenaeus and Alexander were spreading heresy that endangered whole churches. In each case, the danger was more than to the individuals themselves. Furthermore, neither John nor Paul found it to be contradictory to our Lord’s command to privately rebuke those in error.