Former Twitter CEO Dick Costolo told CNBC on Thursday that he's not expecting online privacy concerns to lessen anytime soon, in the wake of reports that federal prosecutors are investigating Facebook's data deals.

"It's just going to increase," he said in a "Squawk Alley" interview, as "people increasingly understand these devices in their homes are listening to them."

Users are still unsure where their data is going, where it's being stored and who has access to it, Costolo said, but added that information is slowly starting to "drip out over time."

"Both that user concern and cross-platform sharing over time is going to further increase pressure on these companies to do something about it," he said. "I don't expect this to end."

Facebook shares were down nearly 2 percent Thursday after The New York Times reported that a grand jury subpoenaed records from at least two device makers that had partnerships with the social network.

Meanwhile, Sen. Elizabeth Warren, who is seeking the 2020 Democratic presidential nomination, wants to break up the massive technology companies and make "big, structural changes" to promote more competition.

But Costolo said it's not going to be easy because lawmakers, during last year's hearings with Twitter CEO Jack Dorsey and Facebook COO Sheryl Sandberg, showed "almost a surprising lack of understanding of how these companies work."

"Companies are very different from each other, you can't just break them up similarly," contended Costolo — a partner at venture capital firm Index Ventures. He said a more-nuanced complaint, like the one that Spotify CEO Daniel Ek levied against Apple with European antitrust regulators, reveals the complexity of a one-size-fits-all approach. Ek claims Apple Music has an "unfair advantage" over rivals such as Spotify.

Silicon Valley executives are still taking note, though. When Republican Sen. Ted Cruz and Warren "agree with each other on Twitter, everyone out here takes notice," Costolo said.

First time I've ever retweeted @ewarren But she's right—Big Tech has way too much power to silence Free Speech. They shouldn't be censoring Warren, or anybody else. A serious threat to our democracy.