I actually had a really good DO shadowing experience recently, though it was brief (about 7 hours in the ED). I think it depends on the doctor's personality, honestly. He was very energetic, pretty young, and an attending, so he was a good teacher. I asked lots of questions about the profession, the patients, etc. There were residents around that were really friendly and in a good mood. There was one med student (DO) that I talked to who was really informative. He even invited me to come back.
There's probably a lot of luck involved in that. I found him through the AOA mentoring program, so he put himself out there and wanted shadowers (shadows?). Honestly, I think the real worth in this type of thing is getting to talk about what day-to-day work is like as a physician, not so much the patient stuff you'll see (unless it's surgery, then that's just awesome). I saw people with flu, stomach aches, rashes, etc., so it wasn't anything spectacular, but I really learned a lot about DO just from talking to the residents and attendings there.
He didn't give me the impression I annoyed him by asking questions. I think it would have been really awkward if I would have just followed him around with my arms folded or something. I think if you look interested and communicate the feeling that what you think they are doing is genuinely cool, you'll leave a good impression.
I don't know what kind of work they expected the OP to do...was there a time they wanted you to fetch something for them and you didn't? seems strange.
Oh yeah, don't forget the good old "how to win friends and influence people" principle - act interested in the person you're talking to, ask the doctor questions about him (what his med school experience was like, what he thinks is the best thing about osteo today, does he ever use omm in his practice? etc.) This really gets good results.
