The גמרא says in the name of רב המנונאthat if a woman tells her husband that he divorced her that she is allowed to remarry since חזקה אין אשה מעיזה פּניה בפני בעלה. The ר"ן says that we pasken like רב המנונא and we even go so far as to say that she gets her כתובה (although perhaps not her תוספת כתובה). There is a משנה in כתובות דף כ"בthat says if a woman says I was married but I got divorced she is believed because of הפּה שאסר הוא הפּה שהתיר. However, if there were witnesses that she was married then she is not believed. According to רב המנונא, you would have to say that the משנה is talking about a case where she didn’t say she was divorced in front of her husband because if she did she would be believed even without witnesses because of the חזקה אין אשה מעיזה פּניה בפני בעלה. Most פּוסקים (רא"ש, ͏ר"ן, ͏רמב"ן, ͏רמב"ם ,רי"ף) seem to assume like רב המנונא and hold that the woman may get remarried with no proof of her divorce whatsoever besides her words. However, the מפרש explains the ספק of the גמרא to be is she believed to stay married to her husband if he is a כהן . The שלמי נדרים asks why he didn’t say like the other ראשונים that the question was whether she could marry someone else? He answers that the מפרש must hold it is פּשוטto the גמרא that she can not marry someone else just based on her words. The ריטב"א here in the name of the רא"ה has a very impassioned piece where he states that “אין הדעת סובלתו” that a אשת איש which is very חמור can get remarried just based on the סברא of אין אשה מעיזה פּניה בפני בעלה, especially when her husband is screaming it’s a lie. It is true we believe a woman to get remarried just because she says her husband is dead but in that case she is afraid that if she is wrong her husband will walk in and her children will be ממזירים. By contrast, in our case, she has nothing to fear. Therefore, he paskens that she cannot get remarried without a גט. Moreover, we don’t even force the husband to give a גט because then every woman who wants a גט will just say “you divorced me” and this will force her husband to have to divorce her. It sounds like the רא"ה did not really accept the סברא of אין אשה מעיזה פּניה בפני בעלה. The רמ"ה brought by the בית יוסף inאבן העזר סימן י"ז סעיף ב says that although the גמרא says אין אשה מעיזה פּניה בפני בעלה , that was only previous generations when people did not have such חוצפּא, but nowadays people will easily be מעיז and the חזקה is no longer valid. The רמ"א brings this as a יש אומרים. The ב"ח maintains that בדיעבד if she married someone else we would let the marriage stand.