Chromium is open source. Chrome is mostly open source. Both send a significant amount of data to either Google or your default search provider by default unless you tell them not to. I primarily use Chromium and I've been through the process of disabling all of its reporting several times - it seems to get easier over time, though following the code that might call out gets significantly more difficult.
I currently have my default search provider set to Bing because I hate searching from the navigation bar anyways and it really annoys me that there's no way to turn it off. I suppose that's the price I pay for using a browser built by a search engine.
I fondly remember the days when Michael Arrington was some corporate puke working for RealNames, a search-from-the-URL-bar concept that everyone outside of RealNames and Microsoft hated. Literally. Everyone.
Google made Mozilla profitable by paying them to be the default search engine. The moment they figured out that worked was probably the last moment browsing was safe anywhere - and the last moment being a Mozilla employee was safe as a long term goal for anyone, oddly enough.
Chrome is Chromium, the only difference is branding and dynamically linked plugins (PDF, flash) which you can disable from about:plugins or delete, then you have identical binary to chromium.
At one point I was trying to figure out how much I would enjoy Chrome, and one test I do with my applications is to have tcpdump in the background running while I run them. Whenever I went to any site, internal, external, or whatever, Chrome phones home to specific google sites. I think the published reason has to do with faster DNS lookups, but when I looked the sites up, they seemed attached to ad-related services. I searched around and could not find any setting to disable this feature. No, "Preferences > Under the Hood > Privacy" has nothing for disabling this feature. I think this phoning home was still there for Incognito mode, but time has left my memory fuzzy on that detail; Incognito mode is useless if you want to actually use cookies to maintain some session anyway, and why should I tell google about what accounts I hold across the net?
For the wiseguy who picks up on that last comment, my preference is Firefox with NoScript, AdBlock, and a disinclination for downloading sex.exe, so no, the common tracking systems do not know much about me. However, my ISP is quite familiar with my habits.
Someone recommended to me Chromium, claiming that it was stripped of this nonsense. So I tried that. Chromium did not phone home in my tests. It also lacked a few nice features that Chrome had, as if it were at least a version behind; I cannot remember what they were, only that at that point, I was sick of the hassle and ditched both pieces of software.
If you are concerned about apps phoning home, just run tcpdump/wireshark/whatever and watch. The extra paranoid will route their connections through a box with these tools.
My tests were within two months ago, so I feel the claim is pretty relevant.
Sorry but your comment is just hand waving. Can you give a specific example of those ad related sites Google "phones" whenever you go to any site on a vanilla Chrome installation?
The mere wire traffic is insufficient to implicate Chrome because most commercial websites use some kind of tracking service which behaves this way. Try the same experiment with the tracking services reported by ghostery[1] filtered out
Update: I checked my systems today and could not get it to reproduce. I do not know if that gives Google Chrome a clean bill of health, but I have no more evidence of a phone home.