I have had obsessions with people before.  They are usually just crushes and I don't do anything odd in terms of pursuing these people, I have never bothered anybody, but thinking about them absorbs my life-- I don't think about anything else, ever, in that time.  Now that I have a boyfriend that doesn't happen anymore, but I had an obsession from 6th to 8th grade, 10th grade to 12th, and then my freshman year of college.  You have to be VERY careful about maintaining your distance from a person when they become an obsession, it gives them a very dangerous position of power over you.  My last person obsession turned out to be a very bad person, and my obsession stopped me from seeing that and I ended up getting assaulted, and I was so brainwashed by them that I didn't understand what had happened to me and he got away with it.
I agree with the person that said it isn't a sexual obsession, it isn't for me either.  It's not even really romantic, though it does stem from a crush of sorts.. it just spins out of control from that and eventually stops having anything to do with wanting a relationship with that person.  It's just like being obsessed with anything else.  I don't want to have sex with my political science textbooks either, but I am obsessed with them.  I know it sounds very strange, but to me a person and a book don't turn out that different once they become an obsession.
THANKFULLY nobody ever knew I had these fixations with them.  I never knew obsession 1-- he was just a classmate, obsession 2 I actually dated but he had no idea how obsessed I was, and obsession 3 had an idea but I am sure he didn't think it was anything behind a normal infatuation.