We do the same thing! Getting rid of as much food leftovers is the only way to keep those rascally raccoons out of your site. After we get through cooking, we open the grate and throw anything burnable into the fire and it gets consumed. The only thing that goes in the garbage bag is cans and aluminum foil.
We burn plastic, paper, food left overs, boxes and bags. The clean up is easy and the kids like to "do the dishes". Make sure you don't throw anything in there that is flammable. Kids can sometimes get carried away with the fun of watching stuff burn.
