SWOSUMike

PMantle said it...it's not really relevant, however, they most likely could never catch a deer. Not a pet, most pets lay around all day and could never just see a deer or smell one and catch it...a strong, woodswise hunting dog up with one, but never would catch one on the run. Plus, hunting dogs aren't usually chasing deer unless they're young.
As far as it not being someone's responsibility like someone said to catch and return a dog...well it's not your responsibility to shoot the dog either, most would get bored and go home soon anyway. Whoever it was that shot a collered beagle...you've got to be kiddin. He'd never have caught that doe. Plus you said the doe had be run a long ways...sounds to me as if the dog ran a deer onto your property from somewhere else, doesn't sound like a problem to me. Focus your efforts on coyotes, which really know how to catch and kill deer. A domestic dog is little threat to the deer hurd.