ALABAMA WMA LAW:

Rule 220-2-.55 Conservation
UNLAWFUL:
(c) To use dogs for stalk hunting of deer or for turkey
hunting. (Leashed dogs may be used for the expressed purpose
of retrieving wounded deer, limited to two (2) dogs for each
tracking event).

Sometime it could work against the shooter to run the dog a mile away and not recover it. Where as if it stayed nearby and still was alive there may be a chance for a second shot nearby. I have seen dogs get a deer up and run almost into the next county. In a lot of cases nobody would see the deer.