This process is slightly different for each of the games.
Generally, you need to go to the challenges page and click the “Instructions” button next to the accepted challenge. This has the user you are challenging, in-game username and details.
You then simply have to copy this and add them to friends in the game and create a game to play with the default settings outlined in each of the pages below for each game.
League of Legends Challenging Guide
Starcraft 2 Challenging Guide
Smite Challenging Guide