When Google Analytics isn't active for a website, can Googlebot (or any other Google entity) get any traffic stats about that website in any other way?
No. Not really.
Google can use Adsense data for some things and of course if you use Google Analytics you have given them some valuable information.
But if you think this bleeds over to search performance, it doesn't.
If you do not use Adsense of Analytics then the only data Google has is search performance and click through rate (CTR) which does effect search performance.
For Google, this is the end of the story. For the sake of education, I will go a bit further.
As for anyone else, there are some tools that exist including crawling any performance stats you post. Search engines do not do this except for search if you allow it. However, there have been cases where people have tried to use this data intelligently. There are cases where this data is useful, but not generally for building performance analysis sites. The data does not exist for enough sites and is far too limited in scope to measure.
Amazon and other tool bar issuers, and this can include Google, can potentially track user behavior. But it is user data and not site data. However, much of this extrapolated data and if you know about Amazon, their numbers and data is pure junk. I do not believe that Google tracks or takes any of this data seriously. From a statistical analysis perspective, it is impossible to gauge a sites performance upon a self-selective group and it is further impossible to extrapolate performance data when you do not know what you do not know. The Amazon model, for example, is horribly flawed and impossible to fix. Google is smart enough not to step in this trap.
Otherwise, if you are asking if a site you build can measure the performance of another site simply through observational methods. Then the answer is a resounding "No."
There are several possible ways that Google could get information about your website:
- Google Analytics: I know you already mentioned this, but I wanted to point out that Google has said they won't use Google Analytics data for ranking purposes.
- Google toolbar: The Google toolbar sends data to Google about every page that the user with it installed visits.
- Firefox and Chrome Safe Browsing: Google runs a service that checks the reputation of websites that you visit. It is enabled in both Firefox and Chrome by default and it sends information about each site visited to Google.
- Other PageRank indicators: Other browser plugins that show the pagerank of a page also send the URL to Google for every page visited.
- From ISPs: ISPs often sell data about which pages their users visit. I don't know that Google buys this data, but it could. In the case of Google Fiber, Google is the ISP.
- From DNS servers - Google runs its own popular DNS servers. Users send queries to these Google servers to get the IP addresses of the sites they are going to use.