I had a public room in a public hospital. Maybe it's because I went to boarding school but I kind of liked sharing, it was nice to have some company outside visiting hours.
When I was in the ante-natal ward and I was freaking out they let DH stay and moved me to a private room - generally I think that if your mental state is such that if you won't cope well alone they will let your partner stay.
Personally I was so tired after each birth that I was perfectly happy to send DH home and crash out, in fact I think I was fast asleep long before they told him to leave.