HomePod Pair - Independent Functions

In answering my own question, I found out (thanks to Apple support phone call) that this problem depends primarily on the settings for AirPlay. When a song is being played on Music on the device that's tied to the HomePod (a specific Apple ID), click on the AirPlay icon to see through what speaker the music is being directed toward. The problem I had was due to one of the HomePods AirPlay settings were set up to be heard on BOTH of our HomePods. Correct the destination signal to only one HomePod at a time, unless you WANT both HomePods to be playing the same song.

Thanks for the clarification. We do have a family subscription to Apple Music, so we are able to stream from each of our Apple ID accounts independently now that things are fixed. I just didn't realize it was my limited understanding of how AirPlay worked that kept us from being able to operate the Home Pods independently from opposite ends of our house.