Why can't I delete pictures I've sent in Google Hangouts in my gallery?
You can delete the photos via the Google+ app or web interface. From the left-hand menu, click Photos. If you haven't turned on auto-upload and don't want to, click Not now to get rid of the prompts. Then, if you're in the app, open the menu that slides in from the left again, and choose Albums from the menu; or in the web interface, click More on the top, then Albums on that menu.

From there you can see the Hangout photos, listed separately from the 'real' photo albums. You can select and delete them to your heart's content. Note that "deleting" a photo from this view doesn't delete it right away: it moves it to the "Trash" or "Bin". To finish them off, you can find that on the same menu you were just on, or just wait for 60 days.