I didn't call all of them, I just looked on their websites. Basically I wanted something with no strings attached, where I could just call up and say, hi, I'm a pre-nursing student, and I'd like to shadow tomorrow in Labor and Delivery. I didn't want to have to go through registering as a volunteer, or an extern, candy striper, etc. I think with UMC I said something the first time about considering working there when I got done with school and wanting to see what it was like, or something like that. Anyway it worked
After that I was used to it so I just called them and set up a day. A tip: do not email them; they never return their emails.
That said, I'm sure that although many hospitals don't have a "policy" regarding shadowing, many might be more than willing to let you do so if you just ask. I did that with the Bethany Birth Center in Phoenix, and they were happy to let me come for a day and shadow. I do highly recommend UMC though. They were very student-friendly and all the nurses were used to and very friendly to students, which is more than you can say of a lot of the Phoenix hospitals
By the way, if you find a Phoenix hospital that lets students shadow, let me know - I'm always looking! I'm closer to Phoenix than to Tucson.