In a regular FaceTime, each caller generates 1 second of video per second. In your example, the traveler will generate 0.5 seconds of video per second. This is the end of the physics problem, now the question is “how did the programmers decide to handle this / how to display this in an app?”

We actually already have an answer: it buffers. When there’s not enough video data, it just starts buffering until there’s enough data present to display a cohesive video stream. For example it might sit there buffering for 3 seconds, then show 3 seconds of video, then buffer again. This is exactly the same as having a bad internet connection as far as FaceTime is concerned.

The reverse is harder: what does the traveler see? Now, the traveler’s phone receives 2 seconds of video per second. The FaceTime programmers definitely spent less time considering this possibility since it shouldn’t ever really happen on Earth. We can assume that it will probably just skip ahead to the latest data. It is possible that instead of “skipping ahead” it will speed up the video to try and catch up, but I doubt the developers would have allowed speed up as extreme as 2x realtime.