The problem might be the gaming computer's HDMI audio output. Did you go to control panel, sound, and set to "Stereo" on gaming computer?
If that don't fix anything, maybe switch audio hookups? Audio out (green) from gaming computer can be plugged into streaming computer's line in (blue), and then maybe use headphone splitter. Or, on streaming computer, control panel, sound, double click speakers, and it might have an option to do lag-free audio pass through if "line in" listed in levels.