We use either straw or shavings. Shavings are preferred, but we can't always buy it, and we have plenty of straw to use in the meantime
Everytime we scrape out the loft and cages, it all goes into the garden. Shavings aren't supposed to go into the garden because they can take away the nitrogen from the soil...but...we do it anyway. Pigeon droppings are VERY rich in nitrogen, so it should level out if nothing else
It's recommended that you compost the droppings and such if you currently have something growing in the garden, because putting in straight in there can "burn" the plants with the high nitrogen. Otherwise, after the harvest is over, we don't worry about composting first
For centuries people over in Europe built huge brick dovecotes for pigeons to live and breed in. The droppings would be collected from the ground and sold as fertilizer, and the squabs could be collected for food. I think this would be a great way to keep the feral population down in the cities...feed the homeless people...fertilize lots of gardens...but that's a different story and a different rant, haha