Scroll down a bit on the subsequent screen, and you’ll see the “Attachments” section near the bottom. This contains all of the recent images and files that’ve been sent in that conversation. Neat!
My friends and I…we have fun.
Tap a picture to view it in a larger size, and then you can swipe through to see the remaining images in that conversation. If you find one you want to save, touch your screen to bring up the available controls and tap the “Share” arrow.
When the share sheet pops up, you can pick your poison—send the image on to someone else, copy it, or save it as I’ve indicated below.
And as one final thing, note that you can do this in Messages on your Mac, too! the brilliant Jeff Gamet went over how to do that in a previous tip. Thanks for that, Jeff!