Joint Account Gift Tax

Situation is a joint account where a second, non spouse was added to the original account (parent/child). The original account holder has contributed most/all of the funding of the account; if the second holder uses the account to make a down-payment in excess of the annual exemption for gifts, with full agreement from the original account holder, does the original account holder now need to file a form 709? Should the second holder record the funds of the down-payment as a gift on the mortgage application, though technically as a joint account holder they also own the funds?