How do the SAW movies achieve dismemberment on screen?

You hide the real body part and saw through a fake one, usually using clothes to your advantage. For arms, use a jacket, have the actors arms behind their arm back inside the jacket and the fake arm in the sleeve. You can use cut aways too to your advantage. Shoot the actor as normal moving the saw to the limb, then cut to close up of the limb (which is now fake) and saw through it. Pork is the closest to human flesh for effect.

When I made a short horror movie, we didn't have the budget or anyone with the expertise to make fake limbs. We got around it by shooting the actor with long sleeve jumper (white for the best effect). Then we prepared the close up shot by wrapping a dog bone (a bone that you get to give a dog, not the bone of a dog) in a belly of pork (normally thrown away by the butcher so can pickup for quite cheap if you ask them nicely). In the long sleeve we placed a garden hose I modified. Then we ran the hose along the bone up to just before the position that was cut. Then sawed away with someone pumping blood from the water hose. I recommend to pump in short intervals for best effect.

It is also handy to have more than 1 of the same item of clothing in case you don't get it right the first time. A mix of 50 percent ketchup and 50 percent ribena concentrate worked well for blood, and didn't smell half bad after a while (the actor had to continue wearing the jumper for a bit afterwards) as we wanted to keep the same one for continuity reasons. If you have access to good editing software, you could always use masking and cgi instead. That option wasn't available in my day. That was part of the fun coming up with the creative solutions. Cut aways is the key. Check out some youtube videos, lots of them lay out the process for how to do this!