I’ll jump in, if I may.
I work for a private ambulance company, and also ride with a volunteer fire company. Most of the private ambo stuff is hospital transfers, dialysis appointments, but we sometimes get priority calls.
Ooh, ooh, I’ll take this one! Most of the time, yes, but all too often, people are just clueless. I’ve run right up on people’s rear ends (their cars!), lights going and siren blaring, and they don’t move. I’ll lay on the airhorn, and they finally look in their mirror, and casually move to the right, as if they have all the time in the world.
OTOH, I’ve pulled up to stoplights, not running priority, just a basic transport or going back after a call, and had cars suddenly pull over and get out of the way. Only when the lights and siren are on, moron! 
As for calls…
I’ve never delivered a baby, but I’d love to. We were transferring an 8-months pregnant woman from the local walk-in clinic to a hospital for vomiting and flu symptoms. The doctor at the clinic assured us she wasn’t in labor. On the trip, I asked a couple of times if she was in any discomfort or if she was having any pain … trying to determine if she was having contractions. She kept saying no, she was fine, no pain. When we got her to the hospital, in her bed in the labor and delivery floor, she turned to the nurse and said, “You know, I’m having these weird rhythmic pains. They come and go about every three or four minutes.” I nearly died.
I’ve had a couple odd laying-on-of-hands experiences like VunderBob described.
Been shot at two times. (well, once we were just in the general area dropping off a patient. My partner said, “Did you hear that? Was that fireworks?”
Cops came by and told us to skedaddle. Didn’t need to tell me twice.)
Been yelled at, threatened, spit on, cursed at, vomited on, kicked and hit numerous times. I’ve had many Alzheimer’s patients threaten to kill me, which is usually kind of funny when it comes from a 100-pound, 80-year-old lady.