Since there's apparently no plugin that does it today, is there an easy way to do it manually?
I use a plugin called Avatar Manager for that function. The author has also done a tutorial on how to write your own avatar manager plugin, but unfortunately it is also a bit outdated and doesn't work anymore. Although the plugin states it is compatible up to 3.6.1, it is still working in 3.8.1. I know at this stage the plugin is maintained as I just recently submitted the Afrikaans translations to the author, and that was included in the last version released.
Images are uploaded via the users profile page if "Anyone can upload" feature is enabled by admin. Images are stored the exact same way as any image uploaded via the media uploader, and images are shown in the the Media section in the back end.