When does a flight require a copilot?
A flight requires pilot &amp; copilot when the aircraft being flown requires it. 

A multi-crew aeroplane is one that requires a flight crew of at least two pilots. One of them is the pilot-in-command (the captain) and the other is the co-pilot (or first officer). All jet air transport aeroplanes and the vast majority of turbine powered air transport aircraft and business jet are multi-crew aeroplanes. 

The definition in Annex 1 — Personnel Licensing states that it is: "an aircraft required to be operated with a co-pilot as specified in the flight manual or by the air operator certificate."

So the secondary answer is no, it cannot be decided by a pilot or an operator, it is a statutory requirement.