A couple of things to try:
I have used a Dremel type tool to cut a slot in the remaining piece and used a screwdriver as normal.
You can obtain a type of drill bit that will drill into the screw and allow you to remove it. I believe it is an Extractor or Extracting Bit. If you tell the clerk at the hardware store what the problem is, they'll know what you mean.
Another alternative is to use a normal drill bit and drill out the center of the screw. This is messy and time consuming and risks damage to the threads of the faucet. Normally these screws are brass, so they are soft enough to drill out. But then you have to pull out all the little thread bits from the edges.
Ans 2 - the best way to do it is an extracting bit as said above, but you first need to drill a small hole in the brass screw. Use a 3/32 bit and then insert the smallest extracting bit which is about the same size. Iuse a 'Tee' handle from a tap and die set on the extractor bit.