A quick follow up to my original post.
I am starting to see synchronization of messages on my primary and virtual DIGITS between my iPhone and iPad apps. I suspect the timing is the issue - sync isn't immediate yet as it is with apps like iMessage, Hangouts and Skype for example. I will check the Chrome browser on my MacBook Air later to see if recent messages are there as well.
What I'm not sure about is whether true synchronization is taking place within the Apple ecosystem yet in terms of read/unread status being automatically updated between devices (e.g. I read a message on my iPhone and it is marked as read on my iPad (including affecting the new message badge icons for the DIGITS app).
Message synchronization is actually pretty tricky given that we're not using just an over the top system like imessage. What you are likely seeing is that your mobile originated messages (sms) take longer for them to sync back to your other devices. And depending on messaging type it might be longer based on the type of network your are on. One thing we do is a bunch of message cleanup so that after a bunch of time we go through and update and check the messaging threads to insure they are updated across all devices.
This thread was the product of our fantastic collaborators during DIGITS Beta! As many terms have changed and bugs have been exterminated since the Beta launched, the information here may no longer be accurate. If you still have a question, please search our active threads or ask a new question in the DIGITS space!