I did this program called
at Columbia in NYC last summer, and it was there- at Columbia's College of Physicians and Surgeons- that the ceremony originated, at least in the States. We got short coats (students "typically" wear short half-coats, physicians wear full long coats), for use in the hospitals during the summer, and we did a little mini-ceremony of sorts. At last count, all but one of the 200+ (?!) allopathic med-schools in the US and Canada do some sort. I don't know about the osteo schools though.
Here is one link that mentions it.
At MCV we did a white coat ceremony, where we had a guest speaker, took the Hippocratic Oath, and were "assisted" into our new short white coats by some of the senior attendings at MCV. It was a very moving ceremony, actually, and I found myself getting teary as I thought about my mom (who had passed away a few yrs earlier), and how she had wanted to be a doctor, and how there I was, I med school.
The standard student coat is a short one, and at most places the residents and attendings wear long ones. At some institutions (MCV being one) it was not that long ago (early-mid 90's) that the residents had to wear short coats as well, and you received the long coat only as a chief resident or as an attending.
What has been interesting to observe is the different traditions at different med schools/hospitals. When I transferred to ETSU for my 3rd yr, I noticed all the students wearing long coats - something I would have never even thought about attempting at MCV. So, of course, I bought two long coats once I started at ETSU!