I've noted the same behavior with the signals 'mouse_entered()', 'mouse_exited()' and 'input_event( Node viewport, InputEvent event, int shape_idx)' on Area2D nodes. It works just fine if the tree is unpaused, but if the tree is paused these signals are not emmited any more, even with the code:
self.get_tree().paused = true
Physics2DServer.set_active(true) # Like Dlean Jeans says
All the nodes are set with the enum value: 'PAUSE_MODE_PROCESS'. We're using Godot 3.1; Can some one help?
In the test we used a button to pause and unpause the tree, and its signals seem to always work fine.
A much weirder behavior occurs when we tried to change the type to StaticBody2D, they seem to "defer" the signals emission to when the tree is unpaused, so a lot of things happen very quickly! Is this as it should be?