I notice Kruetinger in a couple of his videos is able to get a side to side movement with his bottom jaw.   Here's a quick example around 1:32 mark.  https://ru-clip.com/video/Rmbo6peyuJE/ebay-blue-monster-puppet.html

How is this done?

The "hinge" on the mouth plate is loose.  Also pretty sure that there is no real "skull" on this puppet. If any it is just the uppper part of the head that has foam in it.

Thank you, Shawn.

I would say that he must have thumb attached pretty firmly to mouth plate to get that much movement from his thumb in lower jaw.

Yeah , looks like the  'hinge' is pretty loose. There would be quite a gap between the top and  bottom mouth plates, with no attachment between them, but for the  mouth cloth.