I would try to arrange through the physician's office (if s/he has one), by maybe talking to the secretary and asking if the physician is open to shadowers and how to best go about doing it.
The way I arranged my shadowing, though, was through friends or family friends who knew the physician whom I wanted to shadow. They expressed my interest for me, and when I heard positive feedback, I contacted the physician myself.
I shadowed two surgeons (general, ortho) for a day each, and I shadowed a clinical physician (nephro) biweekly for about a year. I'd say the length of time depends on your schedule, the physician, as well as how interested you actually are in that specialty. I didn't go into it planning to shadow for a year - I had a month or so in mind - but the nephrologist enjoyed having me around and I loved the specialty.
Hope this helps! Good luck!