There is no such thing as "private" browsing inside the factory installed browser, with factory installed DNS, and any kind of location data, or other cross-collating information along with your IP. The loss of privacy may be contextual and somewhat statistical, but it would be wrong to assume you weren't identified.
What it does do, is let you see how bidding mechanisms in services like flights and hotels will change bid when the same location as you comes to request service and doesn't have the prior search cookie state. Thats useful I guess.
"find things at a different pricepoint" cookie monster mode?