I have let them have the carcass from Thanksgiving turkeys, as well as the carcass from roasted chicken. They have picked them all clean and been perfectly happy.
I don't do it anymore only because I have several egg buyers who want organic, antibiotic-free, vegetarian-fed etc eggs. So I do what I can to accommodate them. However, in the winter when the eggs slow down and it's just us eating them, I will probably let them pick a carcass again. I like not wasting anything, and they get a lot of enjoyment out of it.
Incidentally, I have found my free range chickens stealing gophers, birds, mice and what-not from my cats. The raw birds don't seem to bother them, either. I think it's gross, but they do it rather gleefully.