It's only starting an engine!
Oh gosh. I must have started an an engine 50-60 thousand times. Think about what you are doing, hit the tit, start the clock, look at the numbers, job done. If it looks wrong, turn it off. As to who starts it, that now depends on what the manufacturer says. Some say, the handling pilot, some say the non-handling, some don't specify. I can't see the problem letting my college in the other seat start an engine all by himself, especially if I am taxiing. He will turn it off it it looks like it's about to cook it itself, he'll have to. The chances of an F/O getting an engine start wrong are considered to be a minor risk that our company is prepared to take. Besides, most F/O's are sharper than me and are probably less likely to get it wrong. On some days I even let them decide on fuel and if they are very lucky, fly the plane.