First I'll give the standard line that you can do a search in this board using "LOR" or "shadow" and find all sorts of past threads about this topic.
Shadowing can take anywhere from just an afternoon to a few months. People who put in a lot of time usually already know a DO and are able to spend more time with them. Finding a DO was the hardest part of my application. I finally found two DO's in two different ways. First I went to some of the schools websites and did some investigation until coming across alumni that were in my area, and field of interest. After sending an e-mail explaining my situtation I got a call back and ultimetly got to spend an afternoon shadowing her in the ED. A few days later she sent a LOR to the schools I was applying to. I met with a second DO by simply stopping in at a local Family Practice and asked to speak to the DO. The registration lady had me write him a note and showed it to him while I waited in the lobby. He said come back the next day at noon, and I spoke with him for about a half hour. Later I shadowed him for two days, three hours each.
It's very awkard to approach someone you don't know and asking for their time and LOR. I'd recommend that when you find someone to meet with, ask about shadowing opportunities and wait until you're done to ask about a LOR.
