/hack
All of these security measures are capable of being circumvented and exploited. The Green Dam software is known to have serious weaknesses and working exploits in the field, essentially making all Chinese computers open to exploit. But the NSA's infrastructure can also be subverted. FBI agents have reported walking into the communications closets where these taps are set up and finding connections and equipment that didn't have the required documentation. Now, the simplest and least nefarious explanation is that some U.S. Intelligence agency is not following the law--unsavory, but not surprising. But a much more alarming explanation is that these taps aren't actually the property of government agencies.
This provides excellent opportunities for hacker characters. Need to get the dirt on somebody with the Green Dam software (or your game world's equivalent), take advantage of the weaknesses and plant your own data-gathering software on their computer. Need to put the frame-up on somebody? Instead of planting spy software on their computer, drop incriminating data on their computer, and then drop a dime on them.
Getting into the communications closets used by the U.S. is slightly harder. You can't just take advantage of weak software, you'll need to compromise physical security. You'll also have more trouble injecting bad data or software into a target computer. The taps are designed for listening only, and altering hardware to allow modifying that data will probably tip folks off that you aren't on the up-and-up. But you would still be able to listen in on any of the communication that you needed to hear.
Getting into a place like this will require decent impersonation skills. This will include the ability to produce verifiable credentials, because there's no chance that the people watching these facilities are just going to let somebody in because they say they're with the government. How you want to play this in your game is up to you. The simplistic answer is to have some variety of secure ID, which of course you can have a good counterfeiter cook up. To make it more difficult, the people securing the facility will have a list of people with pictures who are allowed in, and additions to that list are made by trusted sources.
A slightly more interesting variation would be to have a phone number they can call to verify a person. Given that cell phones with cameras are pretty much the default these days, it is trivial to send along a picture with the phone call. A successful compromise of that kind of system would involve first compromising their communications network, so that their phone calls were routed to a number controlled by your player characters.
You've probably got a lot of other ideas for what you can do with compromised phone and computer systems. I'd love to read your suggestions in the comments below.

