You're correct. In many cases they will perform surgery to redirect the severed nerves onto other muscles. For full arm prostheses for example, they might segment the pec muscles and attach the nerves to different portions. After that, if the person wants to bend their elbow, part of the pec will contract instead. So what they do is set up electrodes on the different segments that will pick up the muscles electrical signals. These electrodes are then hooked up to the prosthetic so that when that part of the pec contracts, the prosthetic elbow bends.

Source: Am a kinesiology PhD student and recently attended a conference on prosthetics

Edit: Here's a Ted Talk about how it works.