Coyotes are a problem, but for years the bears have been coming to gun shots on my farm down east. If you shoot a deer with a bow, you are on the trail immediately. Even with that, I have had a buck taken off my tailgate by bear. I lost 2 bucks in one evening hunt, and had put hands on both of them.
As for coyotes, I predator hunt a lot. Becoming one of my favorite past times with a rifle. As stated above, if a pack comes to the call, I shoot the biggest in the hips to break him down, and his yelps bring the others back out. Can normally drop 2 or 3 more this way. I don't use an AR anymore for coyotes because my trigger finger is faster than my reticle. I find I am times more accurate with a bolt as I know I have to make each shot count.
With that said, I want to control the coyote population not eliminate it, as I feel they have a place in the eco system and I also enjoy the challenge.
Our rule is the same as stated above, any coyote is to be shot while deer hunting, no exceptions.