It would still run afoul of the Constitution, at least by the letter of said document. A bit like the question, "If nobody knows you stole something, did you steal something?" Note that I'm not talking about agents sitting on the street outside your home watching you, or even following you in public. I'm talking about gaining access to your private communications, documents, financial records and so on. And no, said records being under the control of third-parties does not mean they thereby become public, regardless of what government wants to say to the contrary.