Can't recall ever seeing a beagle drag down a deer, although I'll bet somebody shows up here claiming that their Uncle's wife's 3rd cousin has such an incident on video.

In many instances today, people are increasingly being prosecuted for shooting animals that were chasing game but which were not clearly endangering it. In some states, wildlife officials cannot even shoot a dog that is chasing deer until they have first made an effort to catch it live.

So, the shoot and shovel solution can get you in very hot water. Always best to try to work things out another way first. You should also consult the local police, game wardens, etc. about what your specific rights are in terms of being able to legally shoot a dog. Some folks sometimes assume that just because a dog is trespasing or chasing game, it is legal to shoot them. Sometimes those folks are wrong and they end up paying a heavy price for it.

This topic comes up about once a year and usually turns into a ten page war. There is not a clear right/wrong on the issue because the laws and local authorities attitudes toward shooting dogs vary so widely.