When a contract is deployed, it is not executed by the miner. The miner just mines the transaction and makes the byte code of the contract available in the block chain. As a consequence, the creator pays just the cost for contact creation, but not for execution of the contract, and he pays it only to the miner of the block.
A contract is executed if a function of the contract is called. This is again done using a transaction, which is this time directed to the address of the contract address. Now, the contract code is executed, but again only by the miner of the block. The results of the function that was called is put on the blockchain, where it can be read by everyone.
Note that if you are just reading data from the blockchain, but not changing the state of the blockchain, you do not need to send a transaction, and therefore you do not need to spend Ether. In this case, you invoke a function of the contract locally at the node you are connected to, the function is then carried out of the EVM of the node you are connected to.