I think it varies from airline to airline. Most of our (Boeings) only have a tiller on the LHS, so the Captain drives it to the runway then hands over if the RHS is PF. On landing rollout the Captain will take over again and take it to the ramp.
In all cases, once thrust is set the Captain takes or keeps the thrust levers until V1 and initiates any RTO while the PF keeps it straight. Works pretty well.