Not to beat a dead horse, but it didn't work like that last when I gave up on it.
The thing that annoyed me the most it's that when I would pause it for too long it would either disconnect or play without audio or only audio without video. It would usually forget where I was in the video after I restarted the app.
It was enough so infuriating that I deleted the app from my iPhone and iPad after a couple of weeks of this.
It may work well now, but I'm not installing Amazon Prime on one device to stream it to another.