They can either work together, or independently. They can work at exceedingly low speed - like about 100rpm, or anywhere up to full bore - concorde on re-heat mode!
They are not controlled in an 'old skool' manner, but are controlled by the 'Coolant Control Module' (a standard method of operation since the mid 1990s

) - which is now an integrated part of the engine control unit.
VAG-COM, or a stealers 'puter is the only way to
correctly test for correct operation - though if you have a hot engine on tickover, turn on the air con, open bonnet, and watch - they should both be turning.
If either are toast - bin it and buy a new one.
HTH