Are medical doctors actually allowed to be prison executioners?

As someone who's been stuck several times before the person with medical training taps in the next person, who also needs several tries, it is not my experience that you can always find a vein (unless I'm just really unlucky and the hospital employs a lot of incompetent people). Sometimes a nurse can get blood out on the first try, but sometimes they're sticking both arms and moving onto my hands and I end up with two arms covered in bruises the next day. Sometimes they go get a pediatric needle and that helps. Last time, we called in the "IV Team" who used an ultrasound to place my IV. Some of us just have difficult veins.