We moved our first into his own room at 6 months. It was the first time he slept through the night! I wish we could have done it sooner.
With our second, we moved him out after 2 weeks, I think.
The third baby, born in March, had his own room from the very beginning. I slept in there the first few nights, but then went back to my own room and shut both our doors.
I check on the babies frequently until I go to bed, and then again if I get up during the night and they are still sleeping. I only go in if they cry for more than a few minutes. I close both of our doors so I don't hear every little whimper, but that doesn't stop me from hearing their cries when they are awake and ready to eat.
My youngest is almost 4 months and sleep 6-10 hours in a row at night- mostly 8 hours at a time. On the nights he only sleeps 6 hours, I feed him and he goes right back to sleep. He slept 4 hours at a time after being home only a week, and I think having his own room had a lot to do with that. He likes his peace and quiet.