BuddyPress.org

As I stated above, the problem is not with the BuddyPress software, it is with the environment in which the buddypress.org website is hosted. It is a universal truth that websites will not allow you to upload files (like avatars) unless their permissions have been properly configured. In the case of buddypress.org, they are not (and they cannot be at the moment). It sounds like something similar is happening on your server.