Currently, the responsive navigation functions ( updateVisibleNavigationLinks() and updateVisibleNavigationTabs() ) require public and private tabs to be in the #navigation div. Furthermore, it asks them to be siblings for it to work smoothly. To make a long story short, if you wanted to break up private and public tabs into two areas, you have to literally move the HTML as responsive positioning is extremely buggy and absolute positioning breaks the function (as absolute elements never overflow Im guessing) and also remove the publicTabs (and/or privateTabs) class name.
The solution would be to just completely remove the:
Its a small change that would make customizing the navigation area MUCH easier. Feel free to ask questions on this, as I haven't worded it the best.