Private mode does not provide a method for clearing cookies.
Unlike normal firefox browsing mode, private mode cookies do not show up via the show cookies menu under privacy settings in preferences. Unless a new identity is selected or the browser session terminated the cookies will persist. If this concern is not unfounded, then some method of cookie scrubbing compatible with private mode may be necessary to avoid correlation across site visits/exit changes during a single browsing session.
To upload designs, you'll need to enable LFS and have an admin enable hashed storage. More information