I don't see why you'd have to switch your side of the bed if you have a shoulder injury. It seems the simple solution would be to just sleep on the shoulder that's not bothering you or on your back. Why would you need to switch sides to do that? And personally, if my right shoulder was injured I would find it more comfortable to sleep on the right side of the bed so if I laid on my left side I'd be facing away from the middle and get more air. That's just me though.
I think I would swap sides if I was sharing the bed, so that the affected shoulder was on the outside edge...if I was in a great deal of pain, or it was painful to touch I would be worried in case my other half rolled over on it or banged into it...I did that when I had an ankle injury to keep it out of harm's way