The Pet Exchange is where you can safely send one of your pets to another player. You can trade pets at the Pet Trades. You are not allowed to sell your pets or trade them for anything other than pets of similar value.
If the player accepts a pet you send them, they claim full ownership of your pet and they will have to pay the MP fee below. All pet exchanges are final. If you want to remain the owner of a pet and are only lending it to another player, you should use Pet Lending to make sure your pets safely return to you.