Sure. Although that’s just a matter of unplugging your computer from the Internet. Also, at least in theory, Linux isn’t actively leaking all your data into various Cloud services. Microsoft OneDrive and Google Drive are just invitations for the NSA to paw through your file system.
I just can’t imagine how Linux protects you from posting on Facebook.
For the proprietary software, a lot of it is front-doors. Literally just pay-to-prey. Government agencies pay the big data companies to access their warehouses of scrapped data that come directly off their clients’ machines through explicit information harvesting protocols.
That said, it is technically harder to have a covert backdoor in an open source system. But it isn’t impossible, or even particularly impractical, so long as the vulnerability remains reasonably obscure. It would be naive to assume your standard array of linux oses are unassailable.