If a group of people participate in a mining pool and person 1 finds a valid proof of work on their machine what prevents them from submitting this proof of work for themselves and not giving it to the pool to share between the participants?
The very last line of a block effectively contains:
Reward for Solving this Block: Add 12.5BTC to Account for [Mining Pool].
If the miner tried to change that line to read
Reward for Solving this Block: Add 12.5BTC to Account for [My Personal Account].
the altered line would change the block, which would change the hash, and their Proof of Work answer would no longer be valid.
So if a miner solves a block on behalf of the pool, they cannot submit that block for themselves; it will no longer be a valid solution.