ok, what you want to do is doable, but will take some work. We will assume that all pages normally displayed will be teh PRIVATE pages and a subset will be PUBLIC. Before you start, make sure the menus look like you want, the text color background etc.
1) go view a page then view the source of the page. Find the code that starts <div id="menu1"> and copy everything to the ending </div> to a text editor.
2) remove all the PRIVATE pages and sub pages so this only has the PUBLIC pages.
3) change the <div id="menu1"> to <div id="public_menu">
4) add this code (from the <div id="public_menu"> to the </div>) to ATO->Style & edit HEADER AREA->Configure Header Area right after the %pages
at this point, if you visit the site, you will see two page menu bars but the second looks odd.
5) go back and view the page source and copy it all to a text document. delete everything that isn't part of the CSS, then delete any css that doesn't have 'menu1' as part of the selector. I came up with about 107 lines including some blank lines. Doing this will give you all teh CSS styling for teh ID 'menu1'
6) do a global replace of 'menu1' with 'public_menu
7) add all of that to the CSS Inserts
8) add the following to the CSS Inserts
HTML Code:
#menu1 {display:none;}
body.logged-in #menu1 {display:block;}
body.logged-in #public_menu {display:none;}
at this point if you are not logged in, you will only see the public menu, if you are logged in you will see the normal menu not the public menu.
One Cavaet - If someone knows enough to look at teh source of the page, the will see the html for both menu's and can get the page links. If you want to prevent that, then you will have to dig into the code and construct the menu via PHP