If a private room is not availble when you call and you end up having to share, try to get the side of the room that includes the bathroom entrance. I shared a room with DD2 and happened to be placed on the bathroom (and the window) side of the room, and it made things much more private.
My shared room also had a shared phone, and it was at a time when I did not have a cell phone. Answering calls for each other and handing the phone through the curtain to each other was a pain, and it came up a lot. So, I'd rely on a cell phone exclusively in your room if it's a shared phone; don't even give out the room phone number.
I chose a shared room because DD2 was born overseas and I would have had to pay more for a private room, but nobody could tell me how much more, so I chose the shared room that was 100% covered. My DH wasn't planning to stay with me other than the birth, since he had to take care of our older child, so space for him was not a consideration. My first roommate worked out fine and made me glad I'd saved my money. My second roommate had lots of visitors (the room got very hot), and made me glad I was being discharged in 24 hours.
If you get a shared room, you can also see if they have a visitor room on the floor that you could use as an escape valve if the roommate situation is bad. It'll be a pretty poor substitute if you're not feeling well, but if you feel pretty normal, it won't be too bad.
If money is somewhat a consideration, would it break the bank if you or your child had complications and you needed to stay longer, with you presumably paying the private room rate for that extra night or two? Due to jaundice, that is what happened to me with DD2 and my third child, making doubly glad I hadn't taken the private room and the unknown bill.
Advice and commentary on living overseas
DD1 15, DD2 12, and DS 9