The thing is, Google monitors which search results you click, too. Bing's toolbar just does it via browser integration.
Google can and does track clicks from Bing via Google Analytics. Every time you click a result on Bing and land on a page using Google Analytics, Google knows about it, and they record your Bing search terms from the referrer. The same is probably true for pages with Google ads.
What does Chrome have to do with this mode of tracking? Google Analytics will capture a referrer from any browser that includes it, and no browser that I know of displays a "red warning" when including it.
I didn't say you did, Matt. It seems like my post is being misinterpreted, so I'll try to clarify: The parent post was concerned about "the privacy bit". What I'm pointing out is that the same information (search terms in the referrer) is captured by Google Analytics (and by Apache logs, and by advertisers, and by Facebook via "like" buttons, etc, etc). From a privacy standpoint, there's nothing particularly new going on here, IMO.
Google can and does track clicks from Bing via Google Analytics. Every time you click a result on Bing and land on a page using Google Analytics, Google knows about it, and they record your Bing search terms from the referrer. The same is probably true for pages with Google ads.