It's somewhat like this:
When you battle someone without DC (a complete match) the after the match is over, you get a message on whether you want to save this battle or not. If you save it, you can access the Vs. Recorder to view the match.
After this, if you are online while you were accessing the recorder, it will show a blue screen on which you will have an arrow option. Click on that and then on the video you want to publicise.
Then after publicising it on internet you get a code for the battle (over here) which can then be told to other players, who can then view your match as per recorded online by the "Search by code" option.