Pretty sure no matter which libraries you use, you are always searching and checking out from the Overdrive site. Different window dressing form the referring libraries, but you aren't actually on your library's site any longer. I've gone to lapl's overdrive page directly when all of lapl has been down. So not too surprising they can link accts together by email or IP or adobe reader ID.
Not sure how libraries square this with ALA philophies about keeping lending histories private. Hopefully there is something in the contracts stipulating only records of current check outs be kept, but I kinda doubt it.