I've used both methods and prefer to the PF setting the altitude bug when the AP is on. The critical part of the procedure is that both pilots agree on the new altitude not who sets the altitude.
How an airplane handles a level off with discression to a new altitude can affect the timing of the new altitude. On the Boeing you can set the bottom altitude in the altitude preselect and set the intermediate altitude in the MCDU but in the Embraer, CRJ, and Dornier (airplanes I'm familiar with) your best option might be to set the intermediate altitude in the preselect first then set the lower altitude once the airplane has leveled.
It is a lot easier to explain to the PM what you are doing than to explain what you want done and there is less opportunity for communication error.