I would question why the person who owns the checking account wants to make the payments on a loan that they don't owe, but other than that, I don't know of a reason why this could not be done. You just better be sure you document the authorization, including identity verification to make sure someone is not being victimized here.
Sam Jones owes $5,000 to Harry Smith, and Sam wants to pay Harry using an automatic payment from his own checking account to Harry's account, within the same bank. It's subject to Regulation E and will require that you comply with the statement requirements on both accounts. Sam will have to sign a written authorization, of which he will obtain a copy.
The statement on Sam's account will have to reflect the dollar amount and date of each transfer, and the name on the account (not just the number) receiving the funds. Including the transferee name is often a problem for statement rendering.