NSA / Reuters The National Security Agency headquarters building in Fort Meade, Md.

An effort to stoke protests against NSA surveillance has so far generated some noise, but not much signal.

The Stop Watching Us campaign, an umbrella group promoting privacy rights, launched a push this week to drive protests around the U.S. on July 4 in response to what it sees as illegal government surveillance. But it has seen its call-your-congressman campaign overshadowed by the intense debate over immigration reform, congressional sources tell TIME. Public support, at least as measured in call volume, has been muted.

Josh Levy, the Internet Campaign Director at Free Press, says that so far their site has made about 2,000 calls to Congress. With a handful of other organizations already urging their members to join in the effort, Levy estimates that at least 10,000 calls have been sent to Capitol Hill already – a figure he expects to rise as the movement gains momentum. “This going to be a large, slow building movement,” Levy says. “We’re just getting started.”

Rainey Reitman, the Activism Director at the Electronic Frontier Foundation, one of 115 organizations participating in the campaign, defended the slow start as a strategic choice. By staggering the organizations who put out call alerts, Reitman hopes to prevent callers from getting frustrated by busy signals. Her aim is to ultimately generate many calls, but to do so over a sustained period of time. “I think we’ll continue to see a continued ramp up throughout the rest of the week and then early next week,” Reitman says. “And then hopefully, we’ll see Washington really sit up and listen.”

The phone-call barrage is just the first step in the NSA surveillance opposition efforts. The momentum will continue with the newly founded organization called Restore the Fourth – a double entendre that invokes both the nation’s founding and Fourth Amendment privacy rights — devoted to planning protests in more than 100 cities on July 4 against what sponsors see as a spreading surveillance state.

David Segal, the executive director of Internet activist organization Demand Progress, says the grassroots protests will grab lawmakers’ attention even if the phone campaign can’t. “It may take weeks or months,” he says, “but there is going to be increasing outrage about the scope of what’s going on here.”

In the meantime, however, the Stop Watching Us movement has struggled to have its voice heard amid the noise on immigration. “If our phones were being overwhelmed I would help out with taking calls, and that hasn’t happened,” says an aide to Representative Mac Thornberry, a Texas Republican who serves on the House Armed Services Committee and the Permanent Select Committee on Intelligence. “The only thing that we’ve seen a big uptick in would’ve been immigration.”