Is there a way to limit each user's access to their own directory?
Look at these files https://github.com/mikedeboer/jsDAV/tree/master/lib/DAV/plugins/auth
I haven't gotten that far yet but I want to say that there is a method called to authenticate every action before it is performed. You will most likely need to fork jsDAV and create another "auth plugin". You can then add the git path as a dependency in your apps package.json file.
Just ran into this same issue.
I got the basic auth working, but now how do I get the current user from file/directory node so I can filter out the directory/file listings?