I had a private room for laboring and recovery. And since I really did not want to be in a hospital in the first place -"lack of progress"- if I hadn't had a private room, I would have raised a public stink.
No Hospitals or Birth Centers that I know make you share with other women. The Hospitals here are designed to be for precious bonding time with your family.
Methodist in Rochester (Mayo-owned) has shared postpartum rooms. Private rooms are extra. They almost didn't let my DH stay with me the night our son was born 9 weeks early and across town in the NICU, they finally "pretended" not to notice him sleeping on the floor but told him he'd have to leave if I got a roommate in the middle of the night.
Small local hospital for dd1 and dd2- the LDR rooms were all private but the postpartum rooms were technically "doubles". They had 6 postpartum rooms and rarely needed to double mothers up, but they did let you know it was possible and if they did double you up then your partner could not stay overnight.
Larger hospital planned for this next babe- again they have private LDR suites but it's a newer/larger facility and all of their postpartum rooms are private.
If your insurance is causing problems, ask the hospital billing office for assistance. Good luck!