This feature has been left out for a simple reason: a lot of people play
from their mobile phone and adding messaging during battle would
extend a lot the duration of the match.
Do you know that you can send private message to the other players from
the website? Just go to someone's profil, and send a message.
For example, right here, click on my nickname.
Or, go to the ranking pages and search for someone, or go to your own profil (click on your picture) and select one of your last 25 opponents.
To summarize, anywhere you see a player nickname, you can click on it.