We put the dirty straw on the garden or on top of the compost heap, depending on what's there to be fertilised. Some plants find the manure a bit strong straight out of the coop, but TBH we haven't had an issue yet.

Our coop gets cleaned out by DD twice a week (with my supervision). It's pretty grotty by the time it gets done, but they're out free ranging all day, every day, so not too much of an issue really.