... the fact that such a person [having committed fornication or adultery] comes voluntarily to his or her bishop, this confession would go to show that he or she had truly repented. ... if his or her sins are unknown no publicity whatever should be made of them, and it would be in the province of the Bishop to accord to such a person the privilege of being re-baptized for the renewal of his or her covenants, without making a record of the same, and to then say to him or to her, go thy way and sin no more, leaving the party in the hands of the Lord. If the bishop has sufficient reason to believe that a member of his ward is living in a state of transgression, it would be his duty to call the attention of the visiting teachers to such a case with a view to procuring the necessary evidence, and if found to be advisable, to then procure, against him in the regular way under a complaint made by the teachers. But under no circumstance should the bishop give unnecessary publicity to transgression repented of, or unrepented of, neither should he aim to cover up sin of any kind unrepented of.
[Source: First Presidency, Letter to James W. Lesueur, as quoted in Minutes of the Apostles of The Church of Jesus Christ of Latter-day Saints, 1835-1951, Electronic Edition, 2015]