Discernment and judgement

In social media, there are numerous discussions about Christian leaders. Some are criticisms, some are defence, and some are curses on those who speak. Many in the Christian community are confused. At least there are two levels of problems to be handled. First, it is an issue that could be handled by the local church. Second, there are issues that become public affecting multiple congregations. There are biblical principles and examples for both contexts.
1) Local Church: There are people who may commit sin like adultery. First, the elders are supposed to speak in private. Second step is to speak in the presence of two or three elders. Third step is to announce in the church the discipline process. (Matthew 18: 15-17) Four, Paul asked the church elders to take bold steps and if necessary, handover such people to Satan. (I Corinthians 5::5)
2) Public rebuke: Peter was a senior apostle. But he practiced racism (caste-conscious), hence a junior apostle Paul publicly rebuked him. (Galatians 2:11-14) The behaviour could affect all churches as it was done by a leader and not a believer or elder in a congregation. The leader Peter had authority and respect. His attitude would be detrimental for the early church and to church throughout future generations.
3) Public Announcement: Apostle John had to make a public announcement. False teachers should not be given hospitality or ministry opportunities. “If anyone comes to you and does not bring this teaching, do not receive him into your house or give him any greeting, for whoever greets him takes part in his wicked works.” (II John 1: 10,11)
4) Public denouncement: There are some who do not hold to faith and good conscience and are publicly denounced. Paul declared that Hymenaeus and Alexander were handed over to Satan so that they may learn not to blaspheme. (I Timothy 2:20)
God expects believers to discern with the help of the Holy Spirit and Word of God and take appropriate corrective measures locally as well as in a wider sphere.
Do I discern falsehood, false teaching and false teachers?