What do left handed F-16 pilots do about the right side stick?
It does not truly matter what hand the “stick” is in. It matters how you were trained. It matters to what you have become accustomed. 

Case in point...
In general, the pilot in command of a side-by-side pilot aircraft sits in the left seat. This is the seat in which most people begin their initial pilot training. You will typically control the control yoke with your left hand while manipulating the throttle, mixture, propeller control, radios and avionics with your right hand. When you advance to being a Certificated Flight Instructor, you will move to the right seat. In the right seat, you will you will control the control yoke with your right hand while your left hand does everything else. If you go on to become a captain in an airline, you will move back to the left seat.

If the same pilot starts training in a helicopter, the pilot in command seat is typically on the right side. You would reverse again which hand controls what.

Your training for the particular aircraft and position you fill will dictate how you operate. Not with which hand you are more dominant. Both right-handed and left-handed pilots fly their respective aircraft the same way. They train for it.