Nut, a lot depends upon the time of the year when I'm hunting. Let's start with right now. It's hotter than blazes and I'll try and sneak out if I can to call for the camera. I do carry a rifle in case a mangey coyote should come in. Since less than 1% of coyotes that contract mange survive the infestation, I regard this as doing him and all the other coyotes in the area a big favor. Well furred coyotes are allowed to go on about their business. Even if I'm on an ADC project, summer pelts are worthless for anything, including tanning since the hide hasn't thickened enough to anchor the hair.
Fall and winter are fur months. I shoot to kill. I haul my take home and skin them in the comfort of my fur shed. The pelts are frozen for washing/ fleshing/ stretching and drying later. As a taxidermist, I often skin some of them for lifesize mounts or pedestal mounts. My carcasses are disposed of either at the lanfill, or I have a few remote tracts of CRP grass which the owner has posted "No Hunting" and allows me to drag the carcasses out to the middle for the vultures, crows, and possums to feed on. Check with your local Conservation Officer or county landfill ppl to see what the law allows. Either way, do not just dump them over a bridge or in a roadside ditch. I can't count the number of times the local Sheriff has asked me to respond and identify what was reported to them as a satanic ritual dog sacrifice. If you have a furbuyer or trapper in the area you know of, ask them. If you don't want to skin your kill, find a young trapper or hunter that would like them. When I used to trap, I hated dealing with possums, but they were a necessary evil. I found a couple young junior high-aged kids that were just starting out and gave them a half-dozen possums a day. At fifty cents a piece come auction time, they were a lot more excited at the prospect of that extra money than I was about the amount of time and mess I had to deal with. Coyotes on the hoof would be a treasure for someone that age.
I do burry most of them untill the ground freezes up so you cant dig a hole and then I have a carcus hole (that is dug out in the fall) that the rest of the carcases go to till summer once and a while I will hang them from the ranchers fence. He knows that I'm doing my job and it acts as a deturant for other coyotes in the area. thats only on the larger ranches and only on inturnal fences. Bugs and worms got to eat also.
Why? I repect and admire the animal, but homage goes only as far as making sure its death was for some reason- fur, ADC. The number of coyotes we kill each year pales in comparison to those that die from other reasons and none of those get buried.
When I was trapping heavy, I would bring a back-hoe home every fall and dig a 10'X10'X5' hole in the back pasture. Every carcass I couldn't sell, went into the hole. The next spring I'd bring the hoe home again and fill it over.