I've stated out of this one until I could think about various terms. I now feel that "serious" injury is one that keeps one off the mat. "Necessary" is unambiguous. I do not think that any injury that keeps one from training is necessary or even desireable. However, I feel that there should be a realisation from your training that this is a martial art. That, to me, means that there is a level of pain, tiredness, soreness after training etc etc. If you can come off the mat and believe that "yeah, it's very pretty but it'd never really work" you are doing yourself and your chosen art a disservice.
I have had injuries (broken toes mostly)but I have also had the usage of Ukemi to save my life, without training hard I probably would not have been able to fall and save myself. Sorry it's a bit rambling but no, I don't think that serious injuries are necessary.