Officially, they don't need your PII to validate your certificate: "The personal data of the certificate holder does not pass through the gateway, as this is not necessary to verify the digital signature" [1]

That said, I've not actually looked in to how these work... In Berlin at least, it's rare to get an ID check at a restaurant, usually someone just glances at your phone.

[1] https://ec.europa.eu/info/live-work-travel-eu/coronavirus-re...

edit: further: "The CovPassCheck app processes the QR code only in the working memory of the smartphone and does not store any data." [2]

[2] https://digitaler-impfnachweis-app.apps.public.bfarm.de/en/c...