Coyote stole fire and gave it to man, so the did at least one good thing for us. They are the trickster, the Loki. Sometimes to man's benefit sometimes not. Some native cultures believed only a medicine man can kill a coyote. I would shoot one just to see if I have the "juice". I've only seen one while hunting.
I had one come right by while muzzle loader hunting for second season doe on one of the coldest days I ever spent in a tree. It moved like a ghost in the deep snow. The coat seemed extra full and fluffy in the late winter. I cocked the hammer and it made a little click that was barely audible to me. It must have sounded like a gunshot to the yote with the way he took off. I can understand why people hunt them. Predator hunting often presents way more of a challenge. The only unfortunate thing about it is often times the meat from a predator is not worth eating from a taste standpoint.
Lebhuntfish,
Sorry to hear about your dogs. Packs of canines kill in the least humane way I've ever witnessed. Mother nature isn't all flowers and baby bunnies, that's for certain. I would imagine your beagles may have done the same to the yotes if things were flipped and they were the bigger ones. It's unfortunate that coyotes have adopted pets as a favorite food source. I suppose it is a testament to their resourcefulness/cunning, as domesticated animals provide a safer easier meal than a wild animal.
If you do decide to try hunting yotes Punch, work on imitating a cat or dog in distress. I've heard it works much better that the traditional rabbit call.