Those judgments have already been made in the Bible and everyone will have to stand before the Lord Jesus to be judged for the the things done in the body.Deciding the woman was guilty of adultery was the judgment and it was a capital offense in that time, prior to the Savior fulfilling the old and bringing a new covenant. However, to be clear, saying adultery is a sin is not judging. IF a person commits adultery, that person has sinned. The judgment comes from God as He is the ultimate judge and decides what is a sin and what isn't. If anyone has an issue with saying murder, adultery, stealing, etc... is a sin, their issue isn't with the person stating the fact, it's with Heavenly Father.
Exodus 20:14 "You shall not commit adultery."
The question here is about casting stones at other sinners.
According to the NT there are two types of people, sinners who have received the redemptive work Jesus did on the cross, whose sins have been forgiven, and therefore are also merciful to other sinners, and sinners who this is not the case.