Are injuries more common on the left or right side of the body?

Don't know. I suspect that this would often depend on your side preference - i.e. Right handed or left handed. Just a thought.
About the same. Sports that require more use of one limb than the other (such as racket sports, bowling, frisbee, discus throwing, etc.), do likely cause injury more commonly to the dominant arm that is used to hold the sport specific object. The ratio of right and left handedness is about 50:50, so even these types of sports and activities should have an equal mix of right and left side injuries.