No, not quite. If you are a low risk, straightforward case, it is unlikely that you will spend your whole labour in hospital. Much more likely that you will a lot of your labour at home, and the hospital will only let you come in when your contractions are very close together (3 mins in my case). If you go in sooner, and they think you aren't far enough progressed, they will send you home!
In my case, I laboured at home for 7 hours, went to hospital, laboured for another 3 hours in a room by myself (midwife-led birthing centre within a hospital) and then had DD in the room. And I got to stay in the room by myself (with DH) for the next 6 hours until I went home as they didn't need the space, so I didn't have to get transferred to the postnatal ward.
And then when you get to hospital, if they consider you are in far enough advanced labour, a midwife will take you straight to a room by yourself (well, with any birth partner, and a midwife). You really really shouldn't be actually delivering the baby in a room with other women - these kind of cases make the headlines ("overcrowding means women giving birth in corridors"). It would be very very unlikely.
I have heard of people in early labour being in a room waiting to be assessed, but if you in a fit state to just wait around in a room, you probably aren't in advanced labour and should go home!
I would recommend checking with your midwife what your local hospital's policy is (eg come in when your contractions are 5 mins / 3 mins apart). We actually rang the hospital from home at least 2 times before they thought that I sounded advanced enough to come in!
Does this help? Different rules might well apply if you have any particuar medical needs - I think my experience is standard for a normal pregancy and normal labour.