Allegations in Jewish Law
מתני׳ הנטען על השפחה ונשתחררה או על העובדת כוכבים ונתגיירה הרי זה לא יכנוס ואם כנס אין מוציאין מידו הנטען על אשת איש והוציאוה מתחת ידו אע"פ שכנס יוציא:
MISHNA: One suspected by others of engaging in sexual relations with a Canaanite maidservant and she was later set free, or one suspected of relations with a gentile woman and she subsequently converted, may not marry that woman, since this will strengthen the suspicions against him. But if he did marry her, they, the judges of the court, do not remove her from him, i.e., they do not require him to divorce her. With regard to one who is suspected of illicit relations with a married woman and they, the judges of the court, removed her from her husband, i.e., required them to divorce due to this, even if the man suspected of the illicit relations subsequently married her, he must divorce her.
שהיה רבי נחמיה אומר אחד גירי אריות ואחד גירי חלומות ואחד גירי מרדכי ואסתר אינן גרים עד שיתגיירו בזמן הזה
As Rabbi Neḥemya would say: With regard to converts by lions, i.e., forced converts such as the Samaritans [Kutim] described in II Kings (17:24–25); and converts who convert based on their dreams; and converts of the time of Mordecai and Esther described in the verse, “And many from among the peoples of the land became Jews; for the fear of the Jews was fallen upon them” (Esther 8:17); all of these are not converts until they are converted at this present time.
הנטען על אשת איש וכו': אמר רב ובעדים
§ The mishna states that one who was suspected of relations with a married woman may not marry her even after she divorces her husband. Even if they marry without permission, they must divorce. Rav said: This is only in a case when there were witnesses to her infidelity, and because of their testimony the court required her first husband divorce her. However, if her first husband divorced her due to suspicion and rumors but without witnesses, her second husband would not be obligated to divorce her.
אמר רב ששת אמינא כי ניים ושכיב רב אמר להאי שמעתתא דתניא הנטען על אשת איש והוציאוה על ידו ונתגרשה מתחת ידי אחר אם כנס לא יוציא
Rav Sheshet said: I say that when Rav was dozing or sleeping he said that halakha, and it is mistaken. As it is taught in a baraita: With regard to one who was suspected of adultery with a married woman and as a result the court requires her husband to divorce her, and later she married someone else and was then divorced by this other, if the one who had been suspected of illicit relations with her then married her, he need not divorce her.
ה"ד אי דאיכא עדים כי אתא אחר ואפסקיה לקלא מאי הוי אלא לאו דליכא עדים וטעמא דאתא אחר ואפסקיה לקלא הא לאו הכי מפקינן
The Gemara clarifies this: What are the circumstances of this case? If it is referring to a case where there are witnesses to their adultery, when another came and put an end to the rumor of her misconduct by marrying her, what of it? If there were witnesses, the adulterers may never marry each other. Rather, is it not referring to a case where there were no witnesses to the adultery, and the reason she does not have to be divorced from her third husband, with whom she committed adultery while married to her first husband, is specifically because another came and, by marrying her, put an end to the rumor? This implies that were it not so, i.e., had she not married someone else before marrying the man suspected of committing adultery with her, the court would have removed her from him and required them to divorce, even without witnesses to their adultery. This contradicts Rav’s statement above that they must divorce only if there were witnesses to the infidelity.
אמר לך רב הוא הדין דאע"ג דלא אתא אחר ואפסקיה לקלא אי איכא עדים מפקינן אי ליכא עדים לא מפקינן והכי קאמר דאע"ג דאתא אחר ואפסקיה לקלא לכתחלה לא יכנוס
The Gemara responds: Rav could have said to you that the same is true even if another did not come and put an end to the rumor by marrying her. The same principle applies: If there were witnesses to the adultery the court removes her and requires them to divorce, but if there were no witnesses, the court does not remove her. And this is what the baraita is saying: The novelty in this baraita is that even though another came and put an end to the rumor by marrying her, nevertheless, the suspected adulterer may not marry her ab initio due to the original suspicions.
ההוא צורבא מרבנן דהוו סנו שומעניה א"ר יהודה היכי ליעביד לשמתיה צריכי ליה רבנן לא לשמתיה קא מיתחיל שמא דשמיא
There was a certain Torah scholar who gained a bad reputation due to rumors about his conduct. Rav Yehuda said: What should be done? To excommunicate him is not an option. The Sages need him, as he is a great Torah authority. Not to excommunicate him is also not an option, as then the name of Heaven would be desecrated.

What Standard is Neccessary?

א"ל לרבב"ח מידי שמיע לך בהא א"ל הכי א"ר יוחנן מאי דכתיב (מלאכי ב, ז) כי שפתי כהן ישמרו דעת ותורה יבקשו מפיהו כי מלאך ה' צבאות הוא אם דומה הרב למלאך ה' יבקשו תורה מפיו ואם לאו אל יבקשו תורה מפיו
Rav Yehuda said to Rabba bar bar Ḥana: Have you heard anything with regard to this issue? He said to him: Rabbi Yoḥanan said as follows: What is the meaning of that which is written: “For the priest’s lips should keep knowledge, and they should seek Torah at his mouth; for he is a messenger [malakh] of the Lord of hosts” (Malachi 2:7)? This verse teaches: If the teacher is similar to an angel [malakh] of the Lord, then seek Torah from his mouth, but if he is not pure and upright, then do not seek Torah from his mouth; even if he is knowledgeable about Torah, do not learn from him.