First, I'm gonna make one, BIG assumption: You have asbestos, cement shingles. If this is NOT the case, my 'help' may be moot. To be sure, it would be best to have a contractor or someone at a lumber yard or Home Center tell you what you have. Moving right along ...
Yes, it's not out of the ordinary to hear that a contractor will put up new vinyl siding over top of asbestos, cement shingles. Problem is, you MAY void your warranty on the new siding! Also, they're gonna be doing a LOT of hammering so those shingles underneath are going to be loosened up, and begin falling BEHIND your new siding. Now, you'll have a bulging problem. Also, since this type of siding is 'shingled', it's probably uneven. Not only will you be taking the chance of voiding your warranty, you'll have a crappy, uneven looking siding job when it's done. I suggest that either you or the contractor remove the old siding and put up at least a 1/2" insulation board. This way, you've added a bit more insulation and you have a smooth surfact to do the siding on. I'd tape the seams with Tyvek Tape too, or the like.
Now, about whether you should do the shingle removal or not. (... still going with my initial assumption ...) You and some friends / family can do this yourself. Thing is, you should wear long sleeves (like an old white dress shirt so you don't heat up in the sun) and a pair of loose-fitting pants (not shorts), gloves, and an asbestos-filtering mask. Don't get alarmed! These type of shingles become 'dangerous' when you break them, and you'll need to break a LOT of them to have a problem. Nonetheless, treat them ALL with care, and try NOT to break any of them. (Discipline). So you need to be careful and do your best to only break them (while removing them) when it's absolutely necessary. Me, my 68 year old Dad, and my 17 year old nephew removed 2200 sq. ft (22 squares) of the shingles in about 12 hours. I rented scaffolding because it was easier to work. If you use ladders, you'll be doing an aweful log of up-and-down because those shingles are HEAVY. With the scaffolding, I could stack them and move the stacks down, one level at a time. Once I got going, it was a piece of cake. The stacks were put in a good wagon and take to the driveway. If you know someone who's in the 'trades', see if you can hire them for a day or so to help with this, or at least long enough to 'consult' you on the demolition job. This 'person' probably isn't afraid of heights and is familiar with scaffolding and stuff. You see, a contractor can charge anywhere from $75 to $150 per square (100 sq ft) to do the demolition. This is NOT because of the removal and disposal costs (as you will see); it's all labor, and they'd just rather side over the shingles! (My opinion.) I had an experienced contractor (a friend) at $25 per hr (tax-free) help us do the vinly siding part of the job. I did pay him for 2 hours of demolition-consultation work too to get me (us) started, and to show us how to move the scaffolding. Cheap!!!
What we did with the shingles was stack them in the driveway until we were done with not only the demolition but the siding job too. (There was waste from that and the insulation board and stuff too.) Then, I rented a 15 yard dumpster for $425 (I think, maybe less; certainly NOT more). I could have gotten the dumpster sooner but why spend $425 + an extra $75 per week charge for it to sit in the driveway. Besides, I didn't want my neighbors filling it while I slept! I sat it on some 3' long 2 x 6 boards so it wouldn't ruin my driveway. IN less than 2 days from when it was dropped off, I filled 2/3rds of the way! And when my neighbors saw the dumpster, right away, they wanted to know if I'd have extra room for them to throw away some things. (I had already filled it 2/3rds of the way with what I had.) I said they need to pay me $10 for 'a few things'. They were happy and it helped cover the costs a little. You CAN'T fill it past the top. It's the law. The hauler knew what I was putting in it. Fiberglass roofing shingles, asbestos cement shingles, fiberglass insulation, vinly scraps, cinder block, rotted wood, etc., etc., was put in it. Even an old lawn mower. And it only cost me the $425 to haul away. Make sure you got a contract from them stating what's in the dumpster, meaning, it should say 'hazardous' waste, or the like. And, get a reputable hauler! The fly-by-nighters can drive down the street and dump it in the woods if they know they can get away w/it. $425 + $50 (pro help) + $150 (rented scaffolding) + $100 (Nephew labor) + $0 (Dad labor free) = $725 is what it cost me to do the demolition.
When you do the demolition, make sure you do it when there's gonna be a good chance of dry weather. I don't know what you have under your shingles now but if it gets wet, you need to consider that. I worked it such that when my siding friends were gonna come over to put up the 1/2" insulation board, I started my work 3 days before that. Yes, I took off 2 days from work but if you calculate the professional demolition costs at even the low end ($75 x 22 squares=$1650), I was way ahead of the $725 to the demo myself!
I could ramble on and on but this is about the 'jist' of it. IMO, no big deal. If you need more info, feel free to e-mail me directly. My best to ya and hope this helps.