In my outfit the PIC in the left seat has his hand on the thrust levers and makes the decision. If he's being trained in that seat he would still be expected to make the call and do the actions, except that the RHS trainer could also override his decision. Telling him/her that it's NOT his decision would be gross negative training.
If you think about it, it's actually both pilots decision to go flying: I would expect an FO to reject before V1 if a reject scenario presents itself and the captain does nothing.