The hospital where I used to volunteer required quite a few things in order to shadow/volunteer there (they sent all these people through the same process):
Take a class (general hospital rules, what not to touch, HIPAA, etc)
Resume/Statement of Intent
Blood test of some sort
Letter from the Doc you wish to shadow (basically saying they take full responsibility for you)
It's a lot of hoops to jump through, and if you're at a hospital like this the doctors already know what you're having to do to follow them around. At the hospital where I work now, they only require a letter from the doctor and a signed consent from the department. All in all, if you go by the book it could really try your patience.
You are bound to find some doctors who wouldn't mind taking you around under the radar, but, if you expect to do a fair amount of shadowing, jump through the hoops ASAP and you'll have access to any other shadowing opportunity throughout the hospital "free of charge".