OK, well there's a difference between dealing with my own hurt and sadness and someone else's. Sensitive subjects is such a broad category so it really depends on what the subject is and if I have a point of reference. If I have no frame of reference all I can do is listen, comfort, and/or sympathize. If I can relate, besides doing those things I can offer my perspective or how the similar situation affected me.
For example, this may be morbid, but I find it comforting to talk with people who have dealt with cancer or who have a loved one who has had cancer--survived or died. I feel like that is a point of connection between me and that person and if they're willing to go there and talk about it in depth, so am I. Lately, I've found that I'm talking with some of my older coworkers (who haven't pried and been royal pains) about taking care of an sick and elderly parent. That's really something people in my age group (under 30) don't relate to. I've been talking to certain people about death. A couple of people I know have mentioned how layoffs are affecting their sense of self and value as a person, how unsteady and vulnerable this makes them feel.
It depends on context. Like I said, if someone is real, genuine, interested and willing to go there with me, I'll go there as well.
In an intimate situations hurt and sadness will have to be dealt with. My wife will uses fights and arguments in this sense also. We have talked about this and is when she tells me that hurt and sadness is a sign of weakness. Is this why its so important to express yourself in this manner. Not saying its a sign of weakness for you in particular, but a weakness just like its a weakness for me to be able to argue. I can handle being yelled at like you can handle someone crying on your shoulder, just not the other way around.