I put literally everything in my compost piles but wads of grease, which we don't have anyway. We put in dead critters, too, as our compost heaps sit for several years before being used. There may be a random bone now and then but we have never had an issue. No predator troubles, no smell, maybe an occasional mouse nest in a protected corner in the winter but that is it.
Our heaps average 3 by 3 with the big one being about 4 by 5. I don't turn it, the chickens do that, and I will water when I add to the heaps weekly. Some are covered, some are not. It is effortless, unscientific, and a great way to condense all that rabbit/goat/sheep/poultry poo and bedding until it is needed. We live on beyond garbage service and take non recyclables to the dump a couple of times a year. I try very hard fir everything around here to be used at least twice but more is better. Food waste goes to chickens or sheep or the dog then the resultant poo (except for the dog who poos in private) goes in the bin. After a while the compost goes out on the land and helps grow some stuff and we start all over.