Inspiration from elsewhere - 24, CSI

Some good visuals could be perhaps borrowed from the shows like CSI or 24. CSI is infamous by weird lighting and camera angles, 24 proved the concept of showing several movies at one screen when simultaneous events are happening. Not sure how it could be applied to this case.

In case the communication is realtime, both endpoints can be shown in the 24 way.

Another option, when the text is important, may be overlaying the text over the image.

Or the camera can pan across the place, or make details, while the sound of typing serves to keep the continuity of the scene.

Somebody earlier here mentioned the possibility of a character narrating the scene as a voice overlay, describing what's happening instead of letting the audience read the screens. That could actually work pretty well; for multiple-language versions voices are easier to subtitle than screens, and people also vastly differ in their speed of reading text. Would also allow the tricks with camera angles without losing important information.