You can now create a new Multibit wallet, and withdraw the bitcoin from localbitcoins to your newly created address in your Multibit wallet.
I see, but that way I don't own the original wallet, and have to pay fee for the transaction.
Yup, you don't own the address or wallet on localbitcoins, no matter what.
The deposit addresses on exchanges like bitstamp, coinbase, btc-e, localbitcoins are solely controlled by the exchange itself (Exactly as you said, you don't have the access to the private key).
The account holder only has a record of his account balance in the exchange's database.