Katesrider, you can't always be right next to your horse's head when they euthanize it either. If the horse is still up they can leap forward, convulse, etc. It can actually be very dangerous.
We've done both. Dad shot his horse (Red) and he went right down. Flame was already down and couldn't get up. Mom didn't want her shot, so we called the vet. I sat on the ground with her head in my lap when the vet injected her. It was extremely quick and appeared to be very painless. She was ready to go though. It really is personal preference, but people should be aware that a correctly placed shot is painless to the animal.