Okay, I got enough people to unlock the internet tavern. This is pretty interesting.
To supplement the StreetPass stuff, they've implemented a system similar to Bravely Default. Essentially, every 24 hours, you can upload your save data and choose a shard to send out and you'll be able to get up to 3 shards from other folks who have done the same via the internet in return. Sadly, when I did it, I got two duplicates of a shard I already had, but I assume once the game has been out for longer, your chances of getting something better will raise dramatically.
In addition to the above option, you can also download event shards that were uploaded by Square Enix staff. There are two up right now, and both lead to dungeons that are way, way too difficult for me right now. There's also a present code option which I guess allows you to get some other promotional shards via gift codes of some sort.