Very Stupid Apple "Feature" - Shared Text Backgrounds

Your opening post was unambiguous.

It appears that this desired behavior is not part of this still fairly-new feature.

FYI, we’re fellow users who don’t have any inside knowledge of Apple’s “thoughts” during the design process. Like you, we only see the “final product” once it’s released.

Suggest that you might provide feedback to Apple via this mechanism:

Feedback - iPhone - Apple

Albatross587 wrote:

Does anyone know if it is possible not to share iMessage backgrounds with people? I want to change my background for people, but I don't want it to change for them, like a shared wallpaper. I want what I want for the text background and I want them to have what they want. This is so dumb. It seems Apple, being Apple, designed it that way. Exactly what I'd expect from Apple, a really stupid "feature".

Just don't use the feature to change the background in the first place. It's not the end of the world to have a plain background for a Messages thread. Also, "Turn conversation backgrounds off" so that you don't see conversation backgrounds – regardless of what others may be seeing or sharing.

Add backgrounds in Messages on iPhone - Apple Support

Some more info on the feature from the iOS User Guide:

Add backgrounds in Messages on iPhone - Apple Support

You can turn it off in settings - apps - messages

otherwise, the backgrounds are intended to be shared.

When I turn it off it turns off all the backgrounds. I want the background, I just don't want it shared and I don't want theirs to change.

Yeah, this has two big issues. Your friend who put something crazy on the background and finding the image your partner likes. Which both are becoming chores.