Student uploading of images from the HTML editor is a missing feature that Moodle sorely needs. The best approach might be to automatically have a separate upload directory created for every student account in each course. Then, if there are "problem" files the teacher could immediately see where to go.
By default files would be uploaded here, but could not be deleted by students. Though this could be set course-by-course by the teacher. You would of course need the appropriate file type/file size filters.
The upload/link interface can be a subset of the one currently available to teachers, but students would only see the contents of their own upload folder.
As the HTML editor is used in several activities
a global solution, that is, not specific to say NWiki, would be best.
I also like your idea of having an additional common student file sharing folder, then a student can decide whether to upload to their private folder, or the public one, and from which to use an image.