When browsing, you can share other users' images as well as your own! You can share to Facebook, Twitter, Message, Email, Tumblr, Google, and more.
To share an image while browsing a user's Profile, tap the image once to see the detail view. This will reveal the options icon in the top right-hand corner.
Select where you'd like to share this image to.
If you don't see all these options, navigate to your Settings>Social and make sure to have all of these options checked.
For iOS: If you're still unable to share to Facebook, you will need to also make sure you're signed into Facebook within your iPhone settings.