I guess I got lucky. I shadowed a cardiologist for over a year and we rotated through 5 hospitals during his 2 week rotation. He would do clinic for 1 week and then 2 weeks rotating hospitals. I was pretty happy since I got the best of both worlds. I would call him up and tell him I'm in the lobby, he'd come pick me up, and then we went to see patients. I dressed in a lab coat so I guess people thought I was his PA. As for the ENT I'm currently shadowing, I've had to deal with a lot of BS paperwork saying I won't say anything about what I see and that I can only be around the doctor and not wander off.
By the way, I met the cardiologist at a conference he was speaking at and he openly said he allowed shadowing. Since I was the shadowing program coordinator, I obviously got first pick in the draft. For the ENT, I was informed by their manager that all the docs allowed shadowing so I picked the DO (who happens to be the badass of the group). It's a safe bet that usually specialists have hospital privileges.