The native messaging application on your phone isn't likely DIGITS enabled (only those on the latest Samsung Galaxy S6 & 7 are) and thus the messages you send via the DIGITS app can't be returned to your phone. One way to see those messages is to install the DIGITS app on that phone as well and enable the line associated with the SIM. In that way via the DIGITS app you can see the entire thread.
We're working with a lot of OEMs for their new devices to get this functionality built directly into their messaging app so that you won't need the DIGITS app on your main DIGITS that you're trying to replicate to other devices.
Hope that makes sense.
That makes sense, good to know.