Go to any event on the map.
When you choose to start it, you can pick solo or versus. You need at least one other person to do versus- you can add another person by going back to the main menu and in the bottom right corner you'll see- "players list." It will show you people you can invite- those around you on the map, those played with before, those on your friends list, etc. I just tried inviting random people around me and actually got some to join and do the versus events with me.
You can check if they joined at the bottom right corner of the main menu at "manage my group." Once they are in your party, start the event and pick versus instead of solo. Complete this a total of 10 times (doesn't have to be all at once and can be on the same event over and over or a combo of different ones, doesn't matter) and you will have the achievement.