dig a pit bury them biggest thing is keep them from the public eye
There was a contest recently in the area and a ton of coyotes were brought in. Someone that had shot a few left the carcasses in a dumpster. Not only is that a legal issue about proper disposal, but the dumpster belonged to the Veterinarian I bring my dogs to. He knows I hunt so I got an ear full of how upset he was...and rightfully so. The carcasses stunk to heck and back the mouths had jaw blocks and zip ties, (which to a hunter is ok but to the normal population that sees "tortured" animals removed from the dumpster of a veterinarian is an issue), some had torn skin, etc.
There are three animal crematoriums in my area which accept road kill, etc. so if there is one in your area, it might be a good alternative for disposing of a carcass.
I haven't had the good fortune to shoot a lot of coyotes, but the ones I've shot were skinned and the carcasses dragged out to the desert. Some of my friends just skin their kills in the field and bury the carcasses where they were shot.
As it is, there perception is less than great for us. Leaving rotting carcasses in a dumpster only adds to the problem.