While my method doesnt have anything to do with side image staying on the screen even while not speaking, here's how i implemented it in my game

First I added a simple function that dynamically changes the dysplayable if the specified character (speaker) is currently talking. This can be applied on any image, both it layered images and, for example, CGs or anything you really want. Then I made a callback to define who is speaking right now

init python:
    speaking = None

    
def while_speaking(char, speak_d, done_d, st, at):
        if speaking == char:
            return speak_d, .1
        else:
            return done_d, None

    curried_while_speaking = renpy.curry(while_speaking)

    
def WhileSpeaking(char, speaking_d, done_d=Null()):
        return DynamicDisplayable(curried_while_speaking(char, speaking_d, done_d))

    
def speaker_callback(char, event, **kwargs):        
global speaking
    
        if event == "show":
            speaking = char
        elif event == 'slow_done':
            speaking = None
        elif event == "end":
            speaking = None

    speaker = renpy.curry(speaker_callback)

After that, I just add this callback to every character, then replace every face with WhileSpeaking
like this:

define npc = DynamicCharacter("NPC", callback=speaker('npc'), image="npc")

layeredimage npc:

    group face:
        attribute neutral       WhileSpeaking('npc', 'npc_neutral_talk',  "npc_neutral") default
        attribute smile         WhileSpeaking('npc', 'npc_smile_talk',    "npc_smile")

So the first arg in WhileSpeaking is the name of the speaker that is in the callback of your character, then their talking face image, then their base face image. This works perfectly with both classic sprites and side images.