Was the original question related to aeroplanes or helicopters ? Given the poster's name and the type of question I had thought it was a heli question, but most of the answers are about aeroplanes - The risks at startup are different.
With aeroplanes the biggest realistic student risk I can see is not holding the brakes and rolling into another aircraft prop-first; Engine overspeeding on startup is not an issue.
With helicopters the biggest risk is starting with the throttle not fully closed. Without any flywheel or propellor to moderate things, the engine will overspeed so fast it'll make your eyes water. Forgetting to apply the brakes and rolling away isn't an issue on any training helicopter I know of.
So for a helicopter the risk happens extremely quickly but only affects the aircraft being started. For an aeroplane the risk is slightly slower to happen, but affects other aircraft (and maybe people !)
When I learnt to fly helicopters, my instructor sat with me for many many starts and then eventually would let me start on my own. All helicopter schools I know follow that same general pattern, and most have a notice on the door to the tarmac which has big bold writing saying something like "Thottle CLOSED !!" so it's the last thing students see as they go out. The general policy at the aeroplane school where I instruct is that we must be in the aeroplane before the student starts up. And so it is in most aeroplane schools.
Having said all of that, it seems to me that the Pilot in Command carries the can. If it's a dual instruction flight, everything is the instructor's responsibility. If it's a solo flight, then the student is responsible.