I'm not exactly sure how difficult this really is but here's an idea. On the Playstation, when you hide the chat from the stream, there's a brief notification that shows someone has typed in the stream. This is great when you don't want the screen to be filled with everything so you can just focus on the game.
A great addition to this would be a notification on the top right hand corner (if unsnapped) to show new chat messages and notifications briefly (e.g. new viewer just joined, new follower, new subscriber, new donation etc.)
Personally I am a in app streamer and having to turn around to my computer to read chat messages or new notifications can be annoying and stops the game play at points.