I've noticed that the Emoji & Symbols dialog appears, but not next to my text cursor, and it doesn't point to where I'm about to enter a character. This suggests that it cannot determine where my text cursor is. This seems to happen is any application window on any screen is in "Secure Keyboard Entry" mode.
In my case, it was iTerm2. I could either hide the iTerm2 window, or disable "Secure Keyboard Entry" using the main iTerm2 menu (top-left, by the apple logo, when iTerm2 has focus). Then the Emoji & Symbols dialog box appears next to my cursor just as expected, and works again.
LEFT: (BAD) Emoji & Symbols dialog window when Secure Keyboard Entry is enabled in at least one application window.
RIGHT: (GOOD) Emoji & Symbols dialog window without any applications using Secure Keyboard Entry. Note how it "points" at the location it is about to insert characters.

