We have social security numbers here, you don't give those to anyone because you'd risk identity fraud the same as in the US. Health insurance cards are unique but not sensitive information. Verifying them could be done in various ways such as simply having them scanned and submitted.
There's a post a bit earlier that I made mentioning the technology adoption cycle. It's useless to give everyone in Québec some coins, even a smaller amount. It will still be mostly the same people getting them (the early adopters) and that's why targeted giveaways are a smarter way to go about it.
As for distribution, you may be right. Although I think distribution is up to around 8-9% in Iceland and there are reports of people using the coins already to buy stuff. If the news gets out, it'd be more than just early adopters claiming coins. But it could just be Iceland was a fluke there... they did get a lot of press, nationally.
The scan/submit thing I assumed is what you meant for the cards. But my question is, how will you know the card and number is actually valid? It's like... pretend I was a dev in the US and accepted driver licenses from a specific state only. It's not hard at all for someone to just grab an image off the internet, do a bit of photoshop, stick some numbers in .. and I'm a citizen from wherever.
Is there a way for you to verify if a number is a real number or if someone is faking their ID?