Please define "private site info". What do you think Google can get with GA installed, that it cannot get with it not installed.
With GA installed, G now has access to:
* Accurate traffic data for js enabled traffic.
* Browser / geo etc. distribution of your traffic and thus demographic info.
* Accurate bounce rate (not just estimated from bounce back to the serps).
* Accurate referral traffic from non Google sources (i.e. every page / link that send actual real visitors - links from urls that don't send traffic may be devalued); your proportion of Google traffic vs other sources (sites that get little direct referral traffic other than from Google may be devalued) - both of these things are mentioned in Google patents.
* Accurate time on site and page views per session, metrics of user engagement (i.e. is the site sticky and do users stick around?)
* View of paths through site and off the site - if sites that you link out to also have GA installed, they can see the traffic that you refer to them through referrer data and cookies.
* Return (bookmarked) user visits over time - proportion of visitors that bookmark and return to your site later, and what they look at when they come back.
* What parts of your site don't get any direct user clicks i.e. that may be obscured from real visitors that click around, and only exist for search engine benefit.
That's just off the top of my head, I'm sure I could think of a few more things.