I've never heard of non private rooms!!! I couldn't imagine having to bunk with someone else!
In the hospital where I delivered my private delivery room had a bed for me, couch for dh, they brough extra cushions for dd to sleep on, tv, mini fridge, dvd, good food, private bathroom with tub, extra sink in the main room, 2 big windows, and was about 25 ft x 18ft or so in size. A couple hours after delivery I was moved to my recovery room. It was totally private, closet, couch, bed, tv, mini fridge, good food, bathroom with shower, extra sink in the main room, closet, window and about 18ft x 10ft or there abouts. My medicaid paid for everything. I had non-medicaid group health with dd (same hospital) and everything was paid for then as well. I honestly don't think I could handle rooming with another mom.
I'd check out the facilities at every hospital your insurance covers and see if you can find one that doesn't offer anything but private rooms. In my city that is the standard. All 4 hospitals have only private rooms.
ETA: sorry, didn't read replies prior to posting above. Glad you found a loophole!