I can only speak for the hospital that I work for, but here's how ours works (to the best of my knowledge, anyway).
The emergency room docs are employed by a third party company that is contracted by our hospital. The doctors' company does their own billing for physician services while we do our billing for facility fees, supplies used, nurses, etc. While the doctors' company actually does their own billing they do have an income guarantee with our hospital that basically ensures they will make money if the patients we have default on their bills or we have a slow month (yeah right). As for in-network or out-of-network, the hospital negotiates contracts with the insurance companies/networks for that and it's transparent as far as the doctors are concerned.
Answer your questions?