Do you recommend the baby sleeps in a bassinet in your room or is it ok if they sleep in their own room (very close to our room) in a cot?

Red Nose recommends room sharing for the first 6 - 12months.
However, for many families this may not be possible, so close supervision of a sleeping baby is recommended .

Parents are not expected to observe baby constantly.
If baby is sleeping in a separate room check baby regularly to ensure that the baby remains on the back and the head and face remain uncovered
https://rednose.org.au/article/roomsharing