I've discovered that SMS in Chrome has a couple of issues:
1) Perhaps a technological backend issue, but no past message show up at first. This would be a great thing to already have in place.
2) I started a new conversation with someone by sending an SMS. When they responded to it, it started an entirely different conversation. So we are now chatting in that conversation just fine, but the original SMS I sent to them was still sitting off in a separate conversation with no responses. Apparently, after closing and reopening DIGITS, that message was moved to the correct conversation.
3) SMS sends out of order to (at least) Android phones for anything over like 140 characters. This is not an issue when I send texts from my iPhone 7 to Android phones - long text messages will be displayed as long text messages. However, sending a long text message from Chrome sends it in several chunks with (1/3), etc at the end of each chunk. The HTC One seems to be able to fit that into the proper order (but still with the 1/3, 2/3 in the middle of the text), but the Google Pixel just displays it all out of order. I've had this problem before when texting cross-platform from Android phones, and I realize that SMS has a character limitation - but with this next-gen DIGITS messaging thing, I would have thought that had been removed. (Don't most phones just send long messages as MMS these days?)
4) My session is not getting saved in Chrome. Every time I close and reopen Digits, even if I just close and immediately reopen the tab, it takes me to the "turn on lines for this computer" page. I always click "keep me logged in" and register. It then logs me in, and my conversation history is there, but I have to reconnect to Google every single time to get my contacts to sync again. In other words, it "loses" its connection to my Google account, and I have to reauthorize it every time I close the DIGITS tab and reopen it.
Also, is there any official way to give feedback to the Beta anywhere? It seems like that'd be a big button on the digits web app!
So far, I'm excited for the future of the program, but it seems as if the user experience needs to be improved a bit for it to be truly amazing.