My volunteering was definitely not typical, but I have seen one or two other people on SDN mention similar experiences, so it must not be THAT rare (of course SDN is also the home of the "MCAT average is a 35 and anything less than 32 probably requires special explanation even for DO schools" attitude).
I worked closely with physicians, doing assessments prior to the doc seeing each pt. I often translated for the pt (I'm bilingual). The docs would occasionally have me write their orders for them to sign off. Whenever there is a surgical procedure to be done, I'm in there with the doc as his [acting] first assist. When we're not seeing a pt, I get to chill w/ the docs as well as the nurses and other techs. I have dinner w/ the docs every shift.
So, sure, I am working independently of the doc but I am also often there to interpret or taking down his orders. I also prep charts for him, get him anything he needs, help with pt education, etc.
I think it's a difference between "seeing" and "doing" as well. Seeing only teaches you so much if you never DO things. I've gotten to develop my bedside manner by working w/ pts on my own. Many of my physicians have stated that a number of the traits necessary are already present. (I am careful how I say that b/c many here are likely to jump on that and call it blasphemy, but I have actually had quite a few docs with whom I work say things along the lines of "you've really already got many of the skills and traits you need to be a great physician," which sounds odd to say to a post-bacc pre-med, but they have apparently seen things they like. You simply cannot learn much from shadowing beyond things to say in an interview.)