SHOULD WRONGDOERS BE REPRIMANDED IN PUBLIC, OR IN PRIVATE?

Matthew 18:15 Moreover if thy brother shall trespass against thee, go and tell him his fault between thee and him alone: if he shall hear thee, thou hast gained thy brother.(KJV) 1 Timothy 5:19-20  Against an elder receive not an accusation, but before two or three witnesses. Them that sin rebuke before all, that others also may fear.(KJV)

It depends on what the 'wrongdoing' was.  If it's a 'fault' (a personal wrongdoing against you alone), then tell him in private.  If it's something that warrants an 'accusation' (that is, a criminal matter, see the use of the word in John 18:29 among other places), then it should be done before witnesses.  Not a contradiction.