Most pools have an operator. All individual miners part of the pool work on block with address of the coinbase transaction owned by pool operator. pool operator gets all the bitcoins and distributes periodically to individual miners based on the proportion of the work done.
It appears to be me that technically each individual miner can choose the list of transactions except for coinbase transaction. Is this correct? or do pool operators dictate the full block contents? If either way is possible, what usually happens in practice?