I think this depends on the hospital. Here in Dallas, we have lots of hospitals, and most are very proud of their private birthing suites and private post-partem rooms. My hospital has a giant private birthing suite and the post-partem room looks like a swanky hotel (they just did a remodel on them). However, Parkland Hospital in downtown Dallas is where people without private insurance typically go. I've seen a few birthing shows on TV that are set at Parkland, and it appears that their labor/delivery room is shared with multiple other beds separated by curtains.
Do you have private insurance? Have you chosen your hospital yet? What you need to do is call the hospital where you plan to give birth and ask for a tour of the birthing facility. Most hospitals will do this for you at your request and it will give you a much better idea of the facility.