Welp, solved it finally. For anyone finding this in the future, it was something basic and obvious: My other Steam user needs to own the game through Steam. Which is odd, because that’s not true of ShooterGame, and apparently sometimes true of Spacewar (AppID=480) and sometimes not. And sometimes you get an error in the logs like “Could not resolve X to a unique user ID”, but only under very specific conditions. Note also that Steam will happily give you the Steam overlay, let you take screenshots, etc. on a game that you don’t own.
So that’s confusing. You’d think a tutorial would mention something about it.