In my 21 years in the rink, I've seen two career-ending injuries-- a skater with severe nerve damage from crashing into the boards, and a shattered kneecap (again, skater meets boards).
So here's the heirarchy of who gets a say:
If you are in pain do not train full out. You can skate, as in make it around the rink in skates, but if you have, for instance, dislocated your kneecap and it doesn't feel good, for god's sake, don't jump. You probably know the level of pain that feels like injury as opposed to lack of training. Don't let anyone tell you that you're being a wimp, or that you "can't possibly still be in that much pain."
This one's tricky, because most doctors I've talked to don't get skating at all, and athletes in general only moderately better. Thank goodness I managed to find a bone and joint guy who was a skater, because the first one I had was a complete idiot, insisting that since I'd been skating, I must have broken my kneecap (a rare injury, despite the above) rather than tearing the ligament (extremely common). Doctors also often go straight to "quit," not understanding that you may not consider this an option.
If you're uncomfortable with your doctor's advise, or mistrust her expertise, find, as I did, a doctor who gets it, or ask your doctor to recommend a sports injury specialist.
If your doctor has you on painkillers, please do not skate while you are taking them. I once had a parent come up to the judge's booth at an audition to explain that her daughter had fallen on every single jump because she was, essentially, high on painkillers. We were pretty speechless.
With older skaters, parent and skater need to assess the resumption of training together. With younger skaters, it's the parents call in conjunction with medical advice.
I almost didn't even put the coach on the list, because when it comes to injury that has required medical attention, the coach is almost not even part of the question. I have received emails from skaters asking what to do when their coach pressures them to train, even though one, some, or all of the people above aren't ready. Do not let your coach pressure you to train at a level you do not feel you're ready for. If you miss a competition, or a test, or a season, then you miss it. Better than crippling yourself for life.
Edit. From the comments, a great link from L.A. Skate Dad: http://www.stopsportsinjuries.org/figure-skating-injury-prevention.aspx
Have you had a serious injury? How did you resume training?