If you call a hospital volunteering department, they might be able to hook you up.
I've shadowed a doc at one hospital and volunteered at another and both hospitals had specific shadowing programs, that you apply to through the volunteering office. I called up one hospital to see about volunteering and they asked me if I'd be interested in a shadow program. I participated and it was a good experience. Hospitals understand that there are different types of volunteers, i.e. premeds who want experience versus retirees who need something to fill up their time with. Often hospital volunteering departments have programs suited to the aspiring premed.
Not all hospitals have formal shadowing programs though. Another thought--if you start volunteering somewhere you might meet someone whm you could ask to shadow. Also, as you said you can just pick random names out of a phone book. But I'd try calling a few hospital volunteering offices too. Good luck.