Well, in that case I would do my best to infer based on prior knowledge of the potential empathy recipient what kind of response they would prefer and respond accordingly.
As far as what I prefer? I prefer to be left alone, unless I directly ask for otherwise. I don't at all mind being asked if I would like to talk about something, but if I say no I expect that to be respected and not pushed just because someone can't control their curiosity. When I am at the place I feel the need to talk to someone about something or be comforted by others, I have no problem approaching who I deem "safe" to share my feelings with and do so. So basically, the kind of empathy I want is respect. I want my boundaries to be respected, and I don't want to be "made" to feel guilty if I don't happen to feel like sharing why I feel how I do with you. If I don't, I don't, and I am not going to fake it for your benefit when I am the one who is emotionally distraught to begin with. I can't stand it when people try to make it about them in situations like that as if it were other peoples duty to respond to them just the way they want when they want it.
...And this is why I hate it when I accidentally cry/otherwise emote in an obvious way in front of people. Because then (for the majority anyway) they just "have" to know why. And I get seen as a jerk if I just flat out don't want to talk about it at that point, as if I should have just concealed my emotion for their benefit.