Posted by Vulpliks

Plants vs Zombies only has the Local Multiplayer option.
You can run two instances of MelonDS on the same computer and play it (I tried and it works), however, as it only works with the touchscreen inputs, it will be quite hard to play together.

For example, Mario Kart DS supports joystick inputs, so if I plug two joysticks on the same laptop, I can play locally with a friend, no problem.
But in this case, if you want to play against a friend, you would need to use MelonDS on two different computers in the same network and check if you can do the Local Multiplayer between them.
Here and here there are some information about it.

Posted by Vulpliks

As this game does not support WFC I don't think it is possible to achieve this in the Nintendo DS version of the game.
This Versus Mode is supposed to be played by two players close to each other.

Posted by Wifall

You could try using either Hamachi or Radmin, but the DS local communication is highly sensitive to high response times, therefore any ping above 5 to 10ms or so would be too high and the connection will fail.

What I recommend you to do is playing 2 windows in the same PC and using Parsec to play with your friend.