Different vets have different methods of doing it. It's put in an IV catheter here. Sometimes sedating SQ before can cause a lot of irritation. Any needle with any drug can cause screaming.
What people think they see is upsetting, but more than likely this "gasping for air" is a contraction-type movement of the diaphragm (the animal is already gone).
Veterinary assistants are probably legally allowed to euthanize animals in most states if the vet is around. These places just can't afford for a vet to do everything. Right or wrong, it probably won't change anytime soon.