The ability to watch the internet has always been there. The ability to discern the important things from the junk has always been a problem at a mass scale, but the government has answers for those challenges.
Here’s a demonstration of the US government’s capabilities to monitor the public Internet. Former CIA and NSA Director Michael Hayden was on the Acela train between New York and Washington DC, taking press interviews on the phone. Someone nearby overheard the conversation, and started tweeting about it. Within 15 or so minutes, someone somewhere noticed the tweets, and informed someone who knew Hayden. That person called Hayden on his cell phone and, presumably, told him to shut up.