I've trained through a few injuries. The worst was a torn ACL, which I didn't know was torn (nice job doc), until I tore it again so badly I could no longer walk. Had to take a year and a half off after that. Finished a kobudo grading with a broken finger after my drill partner crushed it with his bo, that was fun times. A few split knuckles from bare hand sparring. I've been pretty lucky over all, even the knee ended up healing well.
Most of the time I have to "train through" something its an irritation more than in injury - my shoulders get tweaky/sticky sometimes (from push-ups, heavy weaponry), plantar fasciitis (impact), iliotibial band syndrome (post knee op side-effect that flares up if I slack on the physio), ganglion cysts in my wrists (also from push-ups), swelling to greater or lesser degrees in my knee. Basically your standard grab bag of assorted inflammations that typically affect a martial artist.
As to whether or not to train, I judge that based on a combination of past experience and the advice of the people I trust for that sort of thing. 95% of the time I go to class. I may tone it down a bit while I'm healing, but I go to class. Ice packs, traumeel, and if necessary advil, usually take care of whatever is bothering me. Massage, yoga, physio, depending on the problem.
Drills can be fine if you can trust your partner to respect whatever limitation your injury imposes. Depends on the injury though. If something is broken you shouldn't even be there.