Wait the US doesn't have a centralized/shared medical records service? I never realized that! In France, I suppose like in the UK, we have our "Carte Vitale" ("Life/Vital Card"), which is scanned each visit to a doctor, hospital, pharmacy, etc, each updating your personal database nationally.
It also serves as a link to your insurance provider in each location, so reimbursements are automatic.
There must be some sort of system that shares info some of the time. I signed up for a new doctor (haven't even visited yet) across the country from my old one, and the website of my new doctor's medical center already knows all my historical vaccine dates. I didn't give any of that information over, just my name, birthday, and SSN.
It also serves as a link to your insurance provider in each location, so reimbursements are automatic.