Off-chain transactions have nothing to do with Bitcoin - they are just changing the balances of their customers in their database. Essentially no difference to you average bank account besides that it only works between two users of the same service. Unless they make agreements between different services making them even more similar to your average bank.

You have to completely trust them because these off-chain transactions are neither visible to the public nor agreed on by the majority of the votes nor secured by any cryptographic or proof of work or whatever measures. All trades at Mt. Gox were off-chain transactions...