Depends how lazy the programmer is who wrote the game.
In theory, all you need the network for is to transmit the moves. Using standard notation, you need about three bytes: "Nb3" for example. This can be compressed quite a bit if desired, as there are only 16 pieces per side and only 64 squares (and some of those combinations are illegal, but on the other hand you need to plan for the possibility of pawn promotion resulting in extra pieces); two bytes per move is plenty.
Not sure how many turns there are in an average game, let's pretend it's 50: 50 x 2 players x 2 bytes per move = 200 bytes total.
In practice, nobody will compress a data transmission that's only a few bytes long, so let's assume a prodigal six bytes per move ==> 600 bytes total. Also, the app will probably send some time info back and forth, to keep the game clocks in synch even if the moves take a while to be sent back and forth. Let's assume four extra bytes per move ==> 2 Kb total.
To this you have to add network "housekeeping" - your phone checking that the transmitter it's talking to is still in range and funcitoning properly, for example.
Your game may also be sending tons of unnecessary data. It might "phone home" so the developer can track who's using it, for how long. There may be advertising or subscription tracking.
Only way to know about your particular app is install a metering app on your phone (actually you probably have one already, just go look at it)
that is one hour long on android
I meant 30 minutes live game.
:)