In the public system, you usually share a room with 1 other person (not more). A lot of the times though there isn't anybody there, just an empty bed. In the private system, your costs may be less when you're young (which lures a lot of people into it), but the costs go up rapidly the older you get. When you opt-out of the public system in Germany it's almost impossible to get back in. The public system will never be insolvent, and will always cover any of your illnesses.
There has been a rare case where we've been denied coverage for something cosmetic or trivial, but we've always been given the opportunity to pay out of pocket if we wanted, and usually the cost of everything is a tiny, tiny fraction of what it would be in the states.