Well, an example of an embarrassing fact would be if you were racist. I think embarrassing is a better description than offensive, since i can't think of many things that would be offensive to people on THIS site. But I can think of plenty of facts which are embarrassing or disgusting that I would hesitate to admit that I would think might offend people, but really shouldnt be offensive, and arent very offensive so you wouldnt really post it in the offensive fact thread.
Like.... I have a skin picking compulsion. I will attempt to squeeze out pus from any bump looking thing on my skin. If i see a dark spot at base of a hair i might try to pop the hair, and pretty much always i can get some pus to come out. I am led to believe my compulsion is kind of like body dismorphia and i dont really have zits there, but it certainly isnt helping my delusion that i can make some pus like fluid come out of any place i choose to pop. My picking causes wounds, which i like to pick the scabs off of, and later this causes scarring. So I have very unattractive skin. Looks like freckles that are fadded and leaking.
phobia.