I believe that, by default, anyone can insert images into Community posts; there's no specific setting to allow/disallow it. That said, the allowed extensions for document uploads in HTML editor setting, found in Settings --> Configuration --> Security on the Uploads tab, is enforced for Community posts.

Thanks, but I can't see anywhere to actually upload a file (typically we want users to be able to upload zip files).

And I can't see any settings relating to it (other than the file type filters where you described).

Ah, I understand, Simon. The menu item shown on the left -- with the red circle -- is the "Add a file" tool. You should be able to use that:

 

OK, I see that in your portal, but the buttons don't appear in our portal. How do I configure ours to show these buttons?