As of February 9th, 2015, Share mode has been patched into the 3DS version of Smash Bros.. Go to the Online menu to find it. From here, you can post, view, and download Mii fighters, screenshots, and replays. You can also send and receive them to/from friends. These options are also accessible from the Snapshot Gallery and Replay Gallery in the Vault, and from the Mii Fighter edit menu.
When sharing Snapshots, you can post them to Miiverse instead. This is useful for Smash Bros. players on non-New 3DS systems, as posting to Miiverse was previously impossible on other 3DS systems.