Hello _juliyyya_,
Welcome to Apple Support Communities.
From what I understand, you are inquiring if iMessage and GamePigeon games cost money to use with a friend abroad, correct? I'm here to help you take a look in to that!
When using iMessage (text in blue), this is built into the software on Apple devices. To use iMessage all that is required is an Apple ID, an Apple device, Wi-Fi or cellular data plan. You can always talk to your internet and cellular provider if concerned about data overage charges. For your friend abroad to use iMessage, they would be required to have an Apple ID, an Apple device, Wi-Fi or a cellular data plan as well. Here is an article on more information about using iMessage:
About iMessage and SMS/MMS
As far as GamePigeon, I would encourage you to reach out to their support or developers on any fees that may occur:
How to contact an app developer
Hopefully this helps!
Cheers!