How To Upload A Picture To Imgur Without Sharing To The Public Gallery (An Unofficial Guide) Addendum: The App

Just like on the desktop site, the iOS App has the option to change from Public to Hidden (before you post, not after; if after, you'll have to delete it and try again).

There should be a button you can press in the top left corner that'll allow you to selfishly keep your picture of Fido all to yourself.

Instructions for the Android App are the same...

...so you can withhold pictures of Mittens from us, too.

Et, voila. Oh, and here are Imgur's disclaimers for good measure.