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Can somebody who has committed a particular sin nonetheless rebuke other people for doing it? Or does he have to first stop doing the sin himself?

Bava Batra 60:

ר' ינאי הוה ליה אילן הנוטה לרשות הרבים הוה ההוא גברא דהוה ליה נמי אילן הנוטה לרשות הרבים אתו בני רשות הרבים הוו קא מעכבי עילויה אתא לקמיה דר' ינאי א"ל זיל האידנא ותא למחר בליליא שדר קצייה לההוא דידיה למחר אתא לקמיה א"ל זיל קוץ א"ל הא מר נמי אית ליה א"ל זיל חזי אי קוץ דידי קוץ דידך אי לא קוץ דידי לא תקוץ את מעיקרא מאי סבר ולבסוף מאי סבר מעיקרא סבר ניחא להו לבני רה"ר דיתבי בטוליה כיון דחזא דקא מעכבי שדר קצייה ולימא ליה זיל קוץ דידך והדר אקוץ דידי משום דריש לקיש דאמר (צפניה ב, א) התקוששו וקושו קשוט עצמך ואח"כ קשוט אחרים:‏

R. Jannai had a tree which overhung the public way, and another man also had a tree overhanging the street. Some passers-by objected and he was summoned before R. Jannai. He said to him: Go away now and come again tomorrow. During the night he sent and had his own tree cut down. On the next day the man came back and he told him to go and cut the tree down. He said: But you, Sir, also have one? He replied: Go and see. If mine is cut down, cut yours down, and if mine is not cut down you need not cut yours down. What was R. Jannai's idea at first [when he kept his tree] and afterwards [when he had it cut down]? — At first he thought that passers-by were glad of it because they could sit in its shade, but when he saw that they objected to it he had it cut down. Why did he not say to the man, Go and cut yours down and then I will cut down mine? — In conformity with the maxim of Resh Lakish, who said: [It is written], Hithkosheshu wakoshu trim yourselves and then trim others.

(Translation courtesy of the Soncino Talmud)

According to Rambam's rulings on Teshuva, it is a Mitzvah to publicize one's misdeeds to prevent others from sinning as a part of personal repent. This also ensures that he will not do it again, as required for a proper Teshuvah.

_{Exact quote needed.}