FWIW, in short, I don't recommend covering over the existing siding for a number of reasons I've stated in the attached Follow-up to a post made just yesterday (like yours. It's about 20 posts below yours.) Asbestos siding - safe?. I talk more about this than the question asks.
Since I don't advocate your particular method of installation, I suggest you contact the mfgr. of the new siding to see what they recommend so you don't void the warranty. Make sure you get the person's name in case there's a problem. I have to wonder what's under the existing siding that you can screw into? If it's just, say, gypsum board, how do you know where the studs are? You see, one additional problem I have with your plan is that these shingles have a historical habit of 'moving' and such, thus, allowing water to get behind them. And you know what water can do to the unprotected part of a home. Also, you'll likely have an 'uneven' flow to the new siding by using your method too. Again, in short, check w/the mfgr. to see if they'll warranty their product in your particular situation (and they may), and what fasteners you are to use.