How do we know that water from faucets in public restrooms does not have water from the toilets in it?

Answer 1

In most areas of the world, 'public restrooms' are built to a UBC or building code, which requires water supply lines providing potable water to be protected from back pressure from the components of the system (toilets). Because there are vacuum breakers in the tanks of toilets, and by design in other types of toilets, water in the bowl/tank can not enter the supply line.

Answer 2

The water in the faucets is from the same water supply that feeds the toilets but it is NOT the water that is flushed from the toilets (the same applies to your home).