That's not an accurate way to look at it. There is no "part of the puzzle". Only "the puzzle". The name of the game is to get a hash that starts with a some number of zeros (higher difficulty -> more zeros). In a pool, each participant sends in answers to the puzzle that are of some much easier difficulty. This allows the pool to gauge how much work they are doing and give out rewards in accordance with that.

There are many ways to do pool mining I'll give an intro to the stratum protocol as it's the most used.
With the stratum protocol each worker is given an ExtraNonce1, which is essentially the workers id each worker will have a different ExtraNonce1, and the size of ExtraNonce2, the number of bytes the worker can iterate. Example ExtraNonce1 "08000002" and ExtraNonce2 4. Each worker is also given enough information to build a unique block header: hash of previous block, coinbase1 script, coinbase2 script, list of merkle branches, block version, bits, and time.
Coinbase1 + ExtraNonce1 + ExtraNonce2 + coinbase2 = the coinbase transaction, this is the transaction where the miners reward goes to. The workers receive coinbase1 and coinbase2 they can not change it.
Once the the coinbase transaction is created it is combined with the merkle branches to create a merkle root. At this point the worker has all the components to create a header which can be worked on by incrementing the standard nonce.

Solominers and pools can see the transactions and select the ones they wish to include. They are completely free to chose which and how many to include.