This Name and Photo Sharing feature is proprietary and does not have anything to do with CardDAV directly. I think if you sync your Contacts in your iCloud (with ADP enabled), your Contact info is still unencrypted.
As far as I know, it‘s like the Share Name and Photo feature of iMessage, which is end-to-end encrypted. What algorithm they use, is unclear. I suppose it‘s similar to how Signal profiles work.
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Nextcloud is great. It has AI features, too! Didn’t test these but overall, Nextcloud is a great product for everyone. Remember, however, that you need to use your own hardware to host your server, otherwise you don‘t have any privacy. You should absolutely take all the steps to check system integrity and harden your server. Not only once, but regularly.
It does, it got ported to Pico a few days ago!