Here, in PA, you can hire a private hauler to haul AND dispose of the waste. Depending on how many 'squares' of waste you have (which is measured in 10' x 10' measurements), the hauler may come w/a dump truck or a dumpster. Sometimes, they'll come with a truck/dumpster that's partially full of 'hazardous waste' and top it off w/what you have. That's why you need to measure/estimate the 'size' of your load based on the number of squares you have. (Just give the wall a 'good measure'. If you have a wall that's 32' x 12', which is 384 sq. ft., you have about 4 squares of siding.)
Also, what's important is that 1) you be SURE to tell them you have asbestos cement shingles, and 2) that your Contract is for BOTH hauling AND disposal; not just hauling. If your contract doesn't say 'hauling and proper disposal', ask the hauler if you can put it on there (for peace of mind since he's working with YOUR siding and it is a hazardous material). One of the reasons for #2 is haulers, from time to time, have been known to dump the 'stuff' on an empty lot somewhere. And if the material gets traced back to you, you will definitely have a hard, legal fight on 'hauling and disposal' of a hazardous material.
Just open the YELLOW PAGES and start calling around. My best to ya and hope this helps.