I've shot a bunch of coyotes with a .22 magnum, and several with my .22 magnum Single Six. It's not bad if the shots are close (I've always kept them within 40 yards, closer is better). I like the standard Winchester 40 grain hollowpoint. It just seems to do especially well out of a short pistol barrel. I like the hyper-velocity fodder, like the Winchester Supreme 34 grain hollowpoint and CCI Maxi Mag +V from a rifle. I've never lost a coyote shot with a .22 magnum, but I've always shot them in the head or throat area, and it often does take two shots. I do handgun hunt for coyotes quite a bit, even written some articles about the subject, and personally, I like service loads (like 115 grain +P 9mm hollowpoints) for coyotes out of an accurate pistol. The recoil is mild, and when that round hits a yote', he's down and out.
Even though a coyote is an irritating varmint at times, I think it is irresponsible of any hunter, especially in the not-so-friendly-to-hunters atmosphere we live in today, to say he doesn't mind to wound an animal. I know wounding is part of it sometimes, but no matter what sort of critter it is, we should always care. Not trying to step on toes here, just my thoughts.