Receiving a photo via iMessage: when is it downloaded?
I don't know about the theoretical, but the actual choices that offered in Messages include tapping twice on an image to access the Copy command, or tapping once to open in the image in its own window, in which case the usual arrow in a box appears, leading to a number of saving and sharing options, depending on what you have installed.

On the Mac, messages and photos (or any other attachment) are automatically synced between devices and stored in ~/Library/Messages/Archives and ~/Library/Messages/Attachments.