We have plenty of space outside. I would be looking to compost typical organic waste mostly, vegetable and fruit remains, eggshells, coffee grounds, maybe leaves and stuff from the yard.
A local said they bury theirs with layers of lawn clippings and sawdust. He doesn't do piles because he says critters dig into them. I know we have squirrels, probably possums, and something underground, maybe moles or voles or something. Don't know if there are raccoons around here or not.
It does freeze here, we get some snow but it varies from winter to winter. Gets very hot in the summer.