I'm following Godot's tutorial for pausing a game here. Under the root node for my scene, I have the "main scene" that I want to stop when pausing, and I also have a hidden "pause menu scene" that I show and want to process when pausing. The "pause menu scene" is instanced and has its "Pause -> Mode" property set to "Process". I have a "pause button" that the player can click to pause or unpause the game. Its pause mode is also set to "Process". No other pause modes for any other nodes have been changed from the default "Inherit".
The result is that, while the pause menu does appear, it acts as though it is paused. None of the signals in that sub-scene fire off, nothing animates, no process or _physicsprocess functions are called. I have a feeling that this has something to do with the fact that the pause menu is an instanced scene, since the button the player clicks to pause/unpause the game still processes, but it isn't instanced.