NZ's Ministry of Health has published a specification for the data in the CovidPass QR code. The spec looks pretty good to me; there's probably enough information for anyone to go ahead and implement a verifier app today, and it should only take a few days to put together a bare-bones verifier app. The spec also tells us a lot about how the system will work. I see some confusion/misinformation out there so here are some observations.
The main idea is very simple. You ask the Ministry (probably via the My Covid Record Web site, but possibly in other ways) to generate a statement of the form "<full-name>, <date-of-birth> is considered fully vaccinated". The Ministry computer system checks your records to ensure that they agree you're fully vaccinated, then generates that statement, digitally signs it with the Ministry's private key, and encodes the statement and the signature as a QR code. You can store that code on your phone, or print it out on a piece of paper. Later, you show that QR code to a gatekeeper who wants to check your vaccination status. They scan it with their own app, which decodes the statement, checks that the statement has a valid signature from the Ministry, and if it does, tells the gatekeeper "<full-name>, <date-of-birth> is considered fully vaccinated". To confirm that you're the person the statement is talking about, the gatekeeper will need to check your driver's license or other ID.
If you're not familar with digital signatures, it's important to know that unlike pen-and-paper signatures, altering the statement invalidates the signature and only the Ministry of Health can generate new signatures that verifiers will accept. This is basic "public key crytography" and generally very secure. To generate a fake vaccine certificate someone would have to break into Ministry computer systems, or feed false data into the Ministry database recording them as vaccinated, or find an egregious bug in the verification software. So of course you can easily copy someone else's statement, but if you change the details to match your own details, verifier apps will reject the new statement; a copied statement is only useful if you can pretend to be the person you copied it from.
For privacy: be aware that when you let someone view your QR code, you're telling them your full name and date of birth. They could record that information if they want to (though there may be legislation soon that restricts what they can do with that information). There is no need for a verifier app to notify anyone of these QR code scans, and I would expect the government's app to not notify or record scans. (Hopefully they'll release the source code like they do for CovidTracer.)
As I mentioned above, you don't need a phone to prove you're vaccinated; your code printed on a piece of paper will work fine. Verifiers will need a phone or similar device, but it doesn't have to be connected to the Internet to verify certificates (though the app will need to be updated once in a while). So DoC rangers could scan vaccination certificates at huts for example.
The data in the QR code currently doesn't record which vaccines you have had or when. In fact the Ministry could choose to issue these certificates to people who haven't even been vaccinated, if there's a good reason.
These signed statements have an expiration date on them, so periodically a particular QR code will expire. People using their phones will probably get the new one automatically but if you carry a printed one, you will need to print a new one every so often. This means the Ministry could change the criteria for issuing new certificates (e.g. to require a booster shot) in the future.
I like the way this has been designed. It could perhaps be a bit simpler — I'm not sure using W3C DID is worthwhile — but it's simple enough. By committing to this spec, it will be pretty easy to integrate certificate verification into other apps. People might even be able to implement interesting enhancements like scanning a QR code alongside a drivers license to verify the name and DoB automatically with one action. Let's hope the Ministry's contractors can finish their backend work and verifier app before the end of this month!