Are transactions in a block decided by mining pool operator? My assumptions:

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.

My question:

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?
technically each individual miner can choose the list of transactions
  except for coinbase transaction. Is this correct?


Technically it is possible. In fact no one do it. Pool operator assembles a list of transactions and miners only calculate sha-hashes. They do not know block contents.


  or do pool operators dictate the full block contents?


Yes