This sounds like an overlay problem. do you have Steam overlay enabled?
Steam settings>Ingame>the first 3 enabled, skip the next enable the last one
scroll down to screenshots>disable save an external copy of my screenshots.
Now exit Steam and restart it.
Your screenshots are already saved to your HDD why are you trying saving them again, if you wish to have a copy of them go to the screenshot folder and copy them.

All your screenshots should be available on your hard-drive, inside your Steam folder and then inside your user data folder, it's numbered so you'll need to open that number, which contains more numbers, they are specific to each game, every screenshot you have ever taken should be inside those numbered folders.

I have no idea if they can be recovered though as I've never had a warning sign show up.

Solution:

- Exit Steam
- Go to ../Steam/userdata/<SteamID>\760
- Delete the file screenshots.vdf (but nothing else!)
- Start Steam again
- Issue should be fixed