These achievements require that you finish a single puzzle of each type (casual, standard, and pro) with 3 other players. Initially, only the casual game mode will be available for your created games. You must purchase standard and pro mode with tokens earned in-game in order to create those game modes. However, this is not necessarily needed. You can join another player's game regardless if you purchased those game modes yourself.
When you look to join Public games, you can see the mode of the puzzle right below the title (though almost all games are usually casual). You just need to get into a game with at least three other players of the specific mode you need. Look for the number of players when you are browsing puzzles (the images of green or grey players), but remember more players can still join after you do. Either finish the puzzle or quit immediately. As long as you come back and click on the game (in the My Puzzles tab), the achievements will pop. On the same note, the 3 other players do not need to be there for the completion, or even complete a single word. They just need to have joined the game for a split second.
NOTE: Casual games will be very easy to come by, but standard and pro could prove more challenging as very few players will create those game modes. If you cant find a game of those types, remember you can always create your own after purchasing them from the in-game store for a meager amount of tokens (a few hundred total).