This thread also describes the issue I'm seeing:
Disable Delivery Receipts From 8300?
And there are similar threads on AT&T's forums that don't seem to have a resolution, either.
Delivery Notification for Blackberry Curve
Delivery Notification for Blackberry Curve - RIM BlackBerry - Wireless Forums from AT&T
Delivery Confirmation Messages - Unwanted
Delivery Confirmation Messages - Unwanted - RIM BlackBerry - Wireless Forums from AT&T
One of them suggests a master reset. :-O Would that make any sense? I'm loathe to try it unless there's a possibility it might actually help something.