ISSUE
Under certain conditions, privated games are not private.

OS
Windows 11

STEPS
1. Go to Steam profile - Edit Profile - Privacy Settings - Set Game Details to Private (all running games and any other launched games going forward will run as private).
2. Launch a game (it should be private)
3. From the Library, go to that game's properties - Privacy - Mark as Private (still private)
4. Go to Steam profile - Edit Profile - Privacy Settings - Set Game Details back to Public (the game that is currently running and marked as private is now incorrectly shown to everyone for the duration that it is running)

TEMPORARY SOLUTIONS
1. Launch any other game and exit it, it should correctly apply the private status to the primary game
or
2. Restart the primary game

No where in the description of Mark as Private does it say the status effect requires a restart, nor should it - the privacy status should be applied immediately.

To make matters worse, for the person playing the game that has it marked as private, their chat window shows to them that the game is marked with the hidden symbol, incorrectly reassuring them that their privacy is respected when it actually isn't.