Miners receive transaction and broadcast it to other nodes. Node that solves the puzzle will add that transaction into the block it mined. But as a miner why would I broadcast a transaction with very high transaction fees. Miner would keep the transactions as such to itself and whenever miner get the nounce, miner will include it in it's block and enjoy high transaction fees.
Sure, they might. But the person who made the transaction wants to get it to as many miners as possible. And if it's a high fee transaction, all the miners want to get it. The nature of the Internet is that if Alice wants to get a message to Bill, and Bill wants to get that message from Alice, Charley can't easily stop them.