We analyze a seemingly simple question: When should government share private information that may be useful to terrorists? Policy makers' answer to this question has typically been "it is dangerous to share information that can potentially help terrorists." Unfortunately, this incomplete response has motivated a detrimental increase in the amount of information government keeps private or labels "sensitive but unclassified." We identify two distinct types of private information that are potentially useful to terrorists and identify the range of conditions under which sharing each can enhance counterterrorism efforts. Our results highlight the complex trade-offs policy makers face in deciding how much openness is right in a world where protecting the people from terrorists has become a central duty of government.