I think usually doctors will let you shadow for a week or two. I guess shadow as much as you can with different doctors, but I don't think it's ever long periods of time with any one specific person.
With shadowing I guess go for breadth rather than depth, in order to see the most variety that you can. But this is my opinion, and I bet there will be a lot of people saying otherwise (ie, to try to really immerse yourself in one field).