Hi Chad,
I am also experiencing this issue while trying to be sure I have everything saved before sending my old phone in as a trade in. I think I may have discovered a workaround or additional method so thought I'd share!
First check that your device is signed into iMessage. I had some that would not download and after toggling iMessage off then on again, they completed. Do this in Settings > Messages: turn the iMessage switch On (or Off then On again).
Second, when viewing the image you will see a small banner at the top of the screen with "From: (Contact Name)>". Tap there and it will take you to the message in which that contact sent you the photo/video.
Long Press the photo within that message and you should get an option to Save. This will instantly save the photo/video to your gallery where you can then manage it properly!
So far I've been able to do this to all but one file and that file appeared differently within Messages. It said "Tap To Download", whereas the others showed their file name or image preview. Since it was just a photo, I took a screenshot and transferred that to my computer.
I hope this helps. Good luck!