I do not consider governments to be the real, or even a significant, threat here. My biggest concern is spying by corporations for marketing / advertising / etc. purposes or to enforce 19th century copyright business models (DRM). Government agencies, however, do piggyback in many cases of the existing corporate surveillance, so cutting the large multinational corporations off from the data they greedily crave does have the added benefit of reducing the much lower risk of government surveillance.
A open blockchain makes individuals very vulnerable to spying by corporations.