When people get a text message, it shows up on both their screen and ours.
Needs More Examples - if they're out there.
Text messaging - known to our non-American readers as SMS - is very different from the normal kinds of dialogue seen in TV and film. Speakers are in different locations, like a phone call, but they're communicating in writing. Unlike with e-mail or letters, the writing is usually so short that switching to voice-over would provide some very jarring transitions. Some creators resort to cutting to a shot of the screen, but that hides the setting and character reactions. One solution (rare, but slowly catching on) is to show the text as an animated effect, either with cartoon-style speech bubbles or just text fading onto the screen. Examples: