I think this differs widely from country to country. Here in Sweden our SSN equivalent (personnummer) is public information accessible to anyone by calling calling the register office and today even via various third party online services.

This is the same here in Iceland. We have a very high adoption rates for internet banking and through each bank you have access to the national registry and can find the SSN for everyone (even the prime minister) by searching by name or address.

Companies publish their SSN on their websites.

The key for recipient in intra-country payments is account number plus SSN.

Identity theft is rare. I've always found strange reading about credit card scams in USA where people go through your garbage and get a credit card in your name. Here you'll have to show up in person and identify yourself.

The same is true in Argentina. The list of voters include this (and anyone above 18 must vote). I'm not sure where you'd find info on minors, but I'm pretty sure it's not meant to be secret at all.