Deposit Check Payable to Self-Authorized Signer

Your contract is with the person who opened the account, not the authorized signer. It's impossible to be sure without reading your contract, but I doubt seriously that you would have a right of offset for an item that was not deposited by the account holder. You can and should refuse to accept items for deposit that have not been endorsed by the account holder himself or herself.

Sometimes, people set accounts up this way because the authorized signer is unable to open a bank account in his or her name.