Sharing iCloud Photos downloaded from Messages - where do they go ??
An earlier thread answers go to iCloud settings and make sure Photos and Messages are turned on. My iCloud settings have Photos turned on but "Messages" is not there.
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If you want to sync Messages through iCloud, run the Messages application. Go into Messages > Settings… and click on the iMessage tab. On that tab, there will be a checkbox to "Enable Messages in iCloud." (You may need to make analogous settings on your other Apple devices.)
It does seem odd for the switch not to be in "System Settings", but I'm only reporting what I see …
Even though I have this checkbox turned off, my Mac and iPhone synchronize messages. But my Mac's Message history does not include messages the iPhone sent or received before the Mac was set up. I suspect that If I were to turn on "Enable Messages in iCloud", the two would try to synchronize that "back-history", as well.