I think I would like to know WHO exactly gets the transaction fee?

I mean, If A sends 1 BTC from her wallet to B and it arrives into his wallet, then who gets a transaction fee.

Isn't this the point of a P2P system?

I understand if you use some 3rd party to do the transfers, as they want to make money, in fact it will be their main way of making money.

In my scenario, is there a fee and who gets it?

K