Mod_proxy isn't going to fix your problem. You're going to need to remove all the hostnames from your content from private. So instead of having links like:
<a href="http://private:60001/something/">
They need to look like:
<a href="/something/">
Otherwise you can look into mod_proxy_html, which will actually mangle the page's content (as opposed to just the request) and map the links in the content on private so that they link to public. See this link for some examples.
Essentially, something along the lines of:
ProxyHTMLLogVerbose On
ProxyHTMLLinks a href
ProxyHTMLLinks link href
ProxyHTMLLinks form action
ProxyHTMLLinks script src for
ProxyHTMLLinks img src longdesc usemap
ProxyHTMLLinks iframe src
ProxyHTMLEvents onclick ondblclick onmousedown onmouseup \
onmouseover onmousemove onmouseout onkeypress \
onkeydown onkeyup onfocus onblur onload \
onunload onsubmit onreset onselect onchange
ProxyHTMLURLMap url\(http://private:60001([^\)]*)\) url(http://public:60001$1) Rihe