At my employer, PF does NOT operate reversers! That is a job for PM. That simple fact allows me to pre-brief that PF does not remove their hand from the thrust levers unless they are happy we can stop, AND PM does not select reverse unless they are also happy we can safely stop. Either pilot can call go-around. We both have a very immediate investment, and clear responsibility, in the landing! ONLY when reverse is selected, are we committed to completing the landing! Not before.
Now that is a real mouthful. All this agreeing with each other in a committee decision makes me wonder who is in charge and who wears the can.
First of all, there is a strong risk of confusion when both pilots are handling the controls at the same time. One on the flight controls and the other on engine controls. By that I mean the reverse thrust levers are controls in the sense of directional control. if you don't believe that, then suggest you read the Boeing 737 FCTM on landing on slippery runways in a crosswind where weather cocking after touch down can cause a problem unless careful use of reverse is made. The heading of that particular page is "Reverse Thrust and Crosswind (All engines). A diagram is shown as a way of explanation. The text then states (quote):
"This figure shows a directional control problem during a landing rollout on a slippery runway with a crosswind. As the airplane starts to weathervane into the wind, the reverse side force component adds to the crosswind component and drifts the airplane to the downwind side of the runway. Also, high braking forces reduce the capability of the tires to corner.
To correct back to the centreline, release the brake and reduce reverse thrust to reverse idle. Releasing the brakes increases the tire-cornering capability and contributes to maintaining or regaining directional control. Setting reverse idle reduces the reverse side force component without the requirement to go through a full reverser actuation cycle. Use rudder pedal steering and differential braking as required, to prevent over correcting past the centreline. When directional control is maintained and the airplane is correcting toward the runway centreline, apply maximum braking and symmetrical reverse thrust to stop the airplane. Use of this technique increases the required landing distance." Unquote.
It should be perfectly straight forward for the PF to land the aircraft and apply his own reverse thrust. There is nothing complicated about that. Any directional control issues will be immediately apparent and handled appropriately by the PF including the reverse thrust. No words have to be exchanged between PF and PM.
The possibility of the PM missing or misinterpreting any instructions from the PF if the PM is in charge of the reversers, is minimised. That is because the PF is handling the reversers. The last thing anyone needs is hands in all directions during the landing run with someone on the thrust levers and someone else on the reverse thrust levers.