Don't know about other areas, but around here, almost no one uses straw in stalls anymore. They use shavings. And, it doesn't take as long to clean a stall as you might think. When I was working on the horse farm, I could clean a stall after the horse had been in it all night in about 15 minutes. And, that included dumping the old bedding out, replacing any that needed to be replaced, scrubbing and refilling water buckets and putting fresh hay in the stall.
As to what they do w/it, it depends on the number of horses and the size of the property. If there's large enough fields, the old bedding and manure is normally put on a manure spreader. Then, when that gets filled, the manure is spread on the field. If you don't have one, or don't have large enough fields, the manure is normally piled. If it's composted properly, many people sell it. Others just give it away.
ETA: Chantilly, if you're looking for manure, look on CL. Many people (at least in this area) will advertise it for free there. If you're close enough to me (South Jersey), I'd be more than willing to give you as much as you want!