That isn't 'printing money'. You get a loan, and you make a contract with the bank. Your IOU obligation is on the assets side of the bank, and it's exactly as many numbers their put into your credit account.
But what happens if you want to make a payment to another bank? Mmh? Then they need the reserve currency, the dollar/euro, which only the central bank can print. The money goes from Bank A's account to Bank B's account.