These two questions from the June 9-10 CBS poll
on the NSA surveillance program do an excellent job of summing up why it might be impossible to accurately gauge public opinion on this topic (emphasis added):
"In order to reduce the threat of terrorism, do you approve or disapprove of federal government agencies collecting phone records of ordinary Americans?"
"In order to reduce the threat of terrorism, do you approve or disapprove of federal government agencies collecting phone records of Americans that the government suspects of terrorist activity?"
If you say the NSA is collecting phone records of ordinary Americans, then public opinion is solidly against what the NSA is doing. But if you just change "ordinary Americans" to "Americans the government suspects of terrorist activity," then the public is overwhelmingly in favor.
Polling outcomes are always at least somewhat dependant on poll wording, but on matters of terrorism and civil liberties this problem becomes extreme. At least one academic study had already previously noted this about public opinion on the Patriot Act:
So it’s worth asking—a decade later, what does the public think about the Patriot Act?
It’s hard to say, according to a 2007 study by Samuel J. Best and Monika L. McDermott of the University of Connecticut. In their paper, “Measuring Opinions vs. Non-Opinions—The Case of the USA Patriot Act,” they found that question wording could drastically swing the response in polls about the Patriot Act, with anywhere from 33 percent to 69 percent of respondents indicating support for the legislation. [...]
When given a general description of the Patriot Act—“The USA Patriot Act makes it easier for the federal government to collect information on suspicious Americans in order to reduce the threat of terrorism”—62 percent of respondents supported the legislation. But when given specific information about the act’s provisions for home and library searches, support dropped to 40 percent and 53 percent, respectively.
Ironically, while technological advancements have made it possible for both public and private organizations to simultaneously monitor the communications of millions of people, it appears beyond anyone's reach to accurately measure what millions of people actually think about those surveillance programs.