Me, too.
I can switch my cognitive empathy on or off, but I very rarely experience affective empathy- if that is even what I'm feeling. I tend to think of it more as sympathy, because although I can feel bad for someone, it is never as a result of a shared emotional state, that is somehow contagious. That I can shut off, but not on.
I think empathy can be "trained", but more in the sense of ignoring it. I consider myself empathic, but have learned to detach myself a lot when necessary. If I see a kid fall down and hurt himself, I really don't like it and feel sorry for him. But then there's the part of my life when I really aim to hurt others, and not only I don't feel bad about it, I really enjoy it. I look forward to hurting my adversary, I look for his weak spots and go after them, the worse he gets hurt the better. If I see his nose is bleeding, or his thigh really hurts, I'll try to aim there again and again. Sometimes I cringe when I look back on certain episodes, but in the middle of action I really don't feel bad for the dude.
What about all the injured kids you see on TV? If you laugh when a kid falls, gets hit by something, etc. on tape how come you don't want to do the same while seing a similar event occur on live stage? Is it because you feel like in one situation it's appropriate and in another isn't?