No, I don't think vaccine proof means handing over all your private information.
For example, the EU Digital Vaccine Passport uses a QR code. And each passport has a unique identifier. The scanner would verify that the unique code is valid, but no other information is passed or retained by the scanner.
https://9to5mac.com/2021/07/01/eu-digital-vaccine-passport/
So, no, if implemented properly, none of your private info is given out.
On the contrary, I am all for vaccine passport. For example, I will never consider going on a cruise again unless the cruise line mandates everyone onboard fully vaccinated.