This does not happen often, though I have been in the situation before. Generally, it involves my mother and sister, when one is calling me to complain about the actions (or general personality and world view) of the other. I can usually see both sides of the conflict, it is usually based on something ridiculous and the result of one of them not being able to mind her own business or ignore the other, and I usually point out this fact. This usually ends with them being somewhat irritated with me. Lately, I have begun to simply they change the subject, or to tell them to call back when they have less to complain about. When I do participate, I remain mostly neutral, only conceding valid factual points, but also attempting to see both sides. With most conflicts I believe that the truth lies somewhere in between.
I have found that most conflicts are based on simple misunderstanding and reinforced by pride and strong egos, and are possibly the result of allowing small misunderstandings to go unattended, and irritations to grow and fester. I am guilty of this on occasion. Whatever their cause, I am convinced that there is no conflict which cannot be resolved with calm, rational discourse, as rare as this may be. And, if it cannot, then there are bigger issues at work, possibly irreconcilable ones. This does not apply, of course, to conflicts with no gray area, such as a horrible wrong committed by someone (cheating, violence, etc).
Nevertheless, I do think that is unreasonable for anyone to expect me to get hearted over something that really does not concern me, even if I do concede one side is right or wrong. I certainly wouldn't except anyone to take sides in one of my personal conflicts. I think it would be very bad manners to involve anyone else at all. I definitely wouldn't lose sleep over losing friends/family due to their own irrational behavior.