How can I remove studs and screws from the inside of drywall when installing a pocket door without needing to do paint and patch work on both sides?
I would go ahead and open the wall from the side you are planning on replacing.

Once you open the wall, you might find out that the drywall has been glued to the studs, in which case your best bet would be to make a clean cut and plan on replacing both sides.

If there isn't any glue, you could proceed two ways:


Use a magnet to find each screw and back them out. Done carefully, this would leave a small hole to patch and paint for each screw. Done un-carefully, there will be an ugly chunk of joint compound to patch.
Once you have access to the inside of the wall, you may be able to use a vibratory (or "oscillating") saw to cut the screw heads between the studs and the drywall. The cutting motion on the screws might still cause the paint and joint compound on the screw heads to crack and fall off leaving holes to patch, but you might get lucky and end up not needing to patch and repaint EVERY hole.


Personally, unless there is something unusual about the wall you're trying to save, with special wallpaper, wainscotting, etc., I think it would be quicker to just cut out and replace the drywall on both sides. Drywall and joint compound is cheap; having a professional come out for a patch on both sides of the wall will not be much more than just the one side. Even if you do the drywall yourself, the time spent in trying to save the wall will be longer than just replacing it.