Short version: Find one other person to play with and try not to complete tetrises, but rather three line matches as this will send your opponent two lines at a time giving them enough time to recover before your next line sending onslaught.
Now for the longer version if you want more detail:
In order to 'send lines', you have to complete 2 lines or more in a single drop.
This achievement must be done online and can be done with any number of other players. It is easier to earn this achievement with 3 other players because the lines you send get distributed among your competitors. This way, even if one person fails out quickly, you can still continue to work away on the other two.
You *can* do this with only one other player, but you have to be mindful of how many lines you send. Setting up your grid so you can do multiple tetrises will cause the other player's grid to fill too quickly meaning they won't be able to bring the level down and will fail out before you send twenty. The best thing to do is to make 3 line matches which will send two lines to your opponent. That will make it easy enough for them to clear away the sent lines so they won't fail out too quickly.
Clearing 2 lines will send 1 line to your opponents, but only sending one line at a time in this way will cause the speed of the dropping blocks to go up sooner than with the send two at a time method. It shouldn't make too big a difference if you send one or two in terms of speed since you don't need to play all that long for the achievement, but it's something else to consider.
If you have another method or feel there's something else that should be added, please let me know!
Good luck!