A study out of the University of Vermont in 2004 had a group perform only 1 legged exercises for a period of 8 weeks. The average strength benefit in the UNTRAINED limb was 36% (hamstring muscles) and 58% (quadriceps muscle) of that achieved in the TRAINED limb. You can benefit from exercise without even moving the injured limb!!
Not using this injured arm/leg, can also result in increased scar tissue formation. Basically by not using the body, you are rebuilding a house with a much more fragile foundation. Therefore when stress is applied to that limb or joint (i.e. playing sports), the risk of reinjury is much greater than before.
Considering athletes are some of the world’s toughest population due to a constant barrage of obstacles, I am amazed by the reactions of so many to stop training completely or even avoid using the opposite limb of the injured area.