I'm thinking of that part in the Dark Knight where Batman goes to China (Hong Kong?) to nab that Chinese accountant for all the baddies. Obviously, he has no limits to his jurisdiction (but that was the first time I saw Batman do anything of the sort, and it surprised me). If Batman considers a person a threat, he'll go through all that trouble to a) stop them and b) prove that he can.
Batman has a deep sense of justice. However, what is his stance on such ambiguous violations of privacy? On a related note, Captain America became Nomad(?) during the Nixon era to reflect anti-government sentiment, but as a very America-based superhero, he (as a character) had much more reason to react to the political environment. Batman usually keeps to Gotham and focuses on local issues (Batman Inc. not included).
Basically, would Batman consider Edward Snowden the threat the government says he is, whether for what he's done or for his potential future actions? Would he praise the man as a hero? Would he even care, enough to act on it? I really don't know.