I've trained pilots for a number of airlines (UK based, predominately) and ALL of them had the non-flying pilot (PNF/PM/NHP) action the drills when airborne. Basically, whoever is/was flying the aeroplane continues to do that whilst the non-flyer does the memory drills with every stage "confirmed" by the Pilot Flying (PF/HP) before the action is taken. This includes the fire handle.
On the ground, the majority of those airlines had the Captain doing what he's paid for (managing the flight deck, managing the crew etc.) whilst the FO actions the Fire drill. The Captain will assume the role of PF after a rejected take-off and then identify the nature of the problem (Fire is usually pretty straight forward when it come to identification). Then the Captain will tell the FO which drill to action, allowing the Captain to monitor those drills.
The above was how we did it in the RAF, too (on my particular fleet).
Some companies specify that the FO will do the drill UNmonitored, whilst the Captain liaises with Cabin Crew and ATC.
Having just checked my Boeing 737-400 QRH, it quite clearly states that in the case of a passenger evacuation, the Captain will operate the fire handles - but does not give a specific "Engine Fire on the Ground" drill.
However, the single most important thing is for everyone to know how THEIR company wants it done. That matters more than what the actual SOP is - because unless all crew members know what they are meant to do, there is a chance that something will be missed.