At my current FTO the trend seems to be that if the student is competant, you let them start the engine by themselves and when you get there they throttle back and you climb on board with the engine running. This is usually after they have begun ex 12/13, so you have been on board for quite a few engine starts.
This has some good plus points, the student becomes more familiar with the start procedure and trouble shooting alone without you if they can't get the thing going. Also, it gives you time to finish your coffee and chat up the ops girl a bit longer
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However, if something really went wrong, engine fire, brake failure etc, how would an insurance company view this. Particularly if the student has not yet flown solo, or without medical etc.
Another consideration I have had is that it makes the student think people getting in/out with the engine running is normal, and having read a few accident reports recently of people going in to props on the ground I wonder if this is a good association to have.
Does every school have this solo engine start policy?