What aircraft type? On the bus our SOP's require the captain to initiate the emergency evacuation and do it from memory. The F/O will follow through the checklist to make sure each item has been dealt with. Its not a challenge and response checklist for us. Just get it done and get the final call into ATC to advise them that we are evacuating before the batteries go off. Then we are out of there, either via the ropes or the doors and slides remembering to grab the loadsheet, high viz and any emergency equipment.
If for any reason the captain is incapacitated then of course the F/O would be calling all of the shots. If the cockpit crew cannot be contacted AND the number 1 has concerns about the safety of the cabin then they are allowed to order an emergency evacuation but only after they have exhausted all means of trying to get through to us.