I've always been a fan of letting the person you're talking to dictate the depth of the conversation, if you're the one with the personal information you feel like sharing. There are several topics which I do not personally consider 'secret' and am not really embarrassed to share, but I often refrain from doing so out of respect for the other person's comfort rather than my own. A conversation, whether interview or no, should be a mutual thing, otherwise you risk one party leaving the encounter feeling either unsettled or focusing purely on that one subject. When you do consider that it is an interview, keep in mind that while doctors often have to be able to discuss sensitive topics gracefully and calmly, you don't want the interviewer's overriding impression after you walk out to be "wow, that was a bit awkward" (and you wouldn't want to associate that impression with a doctor either.)
In my case, this means that I plan to find ways to allude to or vaguely describe some of my own personal history if it is required to answer a question, but I won't be throwing all of the details at them right off the bat. It may mean that some of my answers are slightly less specific than they could be, or it may mean that I get followup questions which probe deeper. I will be prepared to answer those followup questions - after all, don't open doors that you aren't ready to walk through - but that way it is the interviewer's choice whether to go more in depth on a touchy subject.