Chess OS X Catalina Game Connected but Moves not Updating
I am seeing the same thing. I have been testing this out locally with two computers and two gamecenter accounts (both on High Sierra).  I see one game that says "Black to Move", while the other says "White To Move". 

I have found one way to force an update is to go to the menu and choose Game -> Close, and then Game -> New -> GameCenter -> Start, and then it will show existing game center games, and you can choose to have your turn. Pretty annoying though.