Try wide knees or legs... I'd start on your knees first... put your hand in the middle of your chest and your body will contort a little doing them because of the angle... but if you do wide knee thing it should help.
either way they are horrible for your rotator cuff... if you want some of the benefit of them and less risk just do pushups with one hand resting on a medicine ball to anchor yourself and keep your arms wide and focus all the weight on the arm on the ground... =]
shoulder issues are no fun (what may not hurt you now could hurt you in 10 years)
JWSD...you ask "Why do them? What do they accomplish?"
I would say that it is definitely a personal goal thing...but they would help you build "functional" strength in your arms (and core) if you're doing them correctly. I suppose, there could be some ego or bragging rights for doing them too...if that is important to you.