What obligation does an outsider (non-family member) have if she or he suspects that there is abuse in another family? Does it matter if the abused party denies the allegation? What if the abused party requests that nothing be done?
This question does not have an easy answer and it certainly depends upon which community you belong (e.g. Hasidic, Modern Orthodox, etc.). As an attorney and an officer of the court (in Pennsylavania and the District of Columbia), I have to first tell you that a witness to a crime has an on-going responsibility to report the crime. Failure to report could subject you to criminal penalties or civil suit. Be that as it may, in many of the Chasidic camps, a witness would be required to ask a rav (most likely the rebbe). The rav then would focus his questions to determine whether the abuser is a continuing threat. If so, then he would authorize you to report the crime to the authorities; he may even order you to do so. But given the Biblical injunction to judge others charitably (Vayikra 19:15), and the reluctance of chassidic scholars to recognize any secular governments as sufficiently fair to permit application of the concept of dina d'malkusa dina, it is very possible he will instruct you to keep it to yourself. Then, if you report the crime without a rebbe's permission, then you might be considered a moser (informer) and subject to shunning or worse. Oy! AFAIK, the non-Chasidic authorities today would not require you to go to a rav before going to the police. They rely, in part, on the position of the Aruch HaShulchan which said that the laws of fair and democratic states should be honored under dina d'malkusa dina. Chasidim reject that: they argue that, because the Aruch HaShulchan gave England and Tsarist Russia as two examples of such "fair" legal systems, the entire ruling is suspect as a censor's edit. Rabbi Yitzhok Breitowitz, shlita, however, makes the interesting point that it is not a censor's edit. He argues that had the censor written the ruling, he would not have mentioned England. He believes that the Aruch HaShulchan believed he would be censored if he only mentioned England and not Tsarist Russia as examples, and that he added Russia in order to avoid problems with the censor. That makes sense to me.
As for your question about reporting a crime when the victim asks you not to, I suspect that you are talking about spousal abuse. When I was in college, I worked on a crisis intervention hotline. I remember taking a call from a woman who needed the number for the women's shelter and having to hang up the phone before I could give her the number. I had this terrible image of the woman being beaten by her jealous husband shortly after hanging up with me. I can't give you a halachic view on this issue. But I would urge you to contact your local chapter of the Jewish Coalition Against Domestic Abuse (JCADA) for more practical assistance. You should also discuss this with your rabbi.