I have successfully sharpened large scissors with a small belt sander and polishing wheel. The problem with this method for small fly-tying scissors would be the likelyhood of overheating them and ruining the temper of the steel. If the steel turns blue you have gone too far. You might try hand sharpening them on an arkansas stone. Dismantle the scissors if possible and hone the bottom edge of the blade by dragging it crosswise towards you with the inside of the blade away from you. Try to preserve the original bevel angle. I would probably just buy a new set of scissors though.
You may, as Blacksmith suggested, try to dismantle the scissors and then sharpen them on the finest (as in smooth) whetstone that you can find. Many pairs of scissors do not come apart easily so I just open them as far as they will go and sharpen them as he described. Sharpening is not difficult and if you can sharpen a penknife blade you can sharpen scissors. The only trick to sharpening scissors is to go slowly and use a really fine whetstone. In my experience it is rarely necessary to discard scissors once that they are dull unless the blades have been bent or really badly nicked.
I sharpen all my edged items with ceramic rods of varying grit. Scissors don't require much. I run them down the rod holding parallel to the existing edge. This is not the edge where the two halves mate, but is the edge where the two halves pass each other creating the cutting action. It is such an easy task, just a few passes and they are done.