So no, that won't be possible. You cannot really block the screen (or portions of the screen) in a Godot app, from what I understand. Even with OS libraries, you can only really block the entire app (used for DRM or copyright protection) so that's not useful either.

What you could do is have two separate windows for the app. One window would be the one the streamer can record, and the other would be their private window (they can drag this onto another monitor, or just put it in the corner). Sadly, Godot 3.x does not support multiple windows. It looks like Godot 4.0 will, but I'll have to check if that works yet.

If you want to try this with the current stable versions of Godot 3.4.4, you'd need to use networking. So you'd be creating two apps. One is the actual game, and the other is just a small app to display private text or messages. See the docs here for how to network two apps.

https://docs.godotengine.org/en/stable/tutorials/networking/high_level_multiplayer.html