How to mount screw in a worn out hole
Your putty (or better, wood filler) approach will kind of work, but that isn't a good solution for holding screws in wood.  Those products tend to crumble and don't bond all that well with the wood, so they don't hold up well for this use. 

BillDOe's dowel solution is a standard way to get strength roughly equivalent to the original wood.  There's a simpler version of the solution.  Fill the old holes with glued wooden toothpicks.  

You break the toothpicks in half and alternate fat side or thin side in.  Dip the toothpick in wood glue, and stick it in the hole.  Add more until you've filled the hole and can't push any more in.  

Find a nail that's a slightly smaller diameter than the body of the screw (the core under the threads), coat it with a little Vaseline, and tap it into the middle of the toothpicks while the glue is still wet.  This will compress the toothpicks into a solid mass and create a pilot hole for the screw.  

When the glue is dry, remove the nail and trim off the toothpick ends sticking out from the surface.