PRISM doesn't work the way you've described it. It ingests data from the FBI's Data Intercept Technology Unit, which handles court ordered wiretaps on individual accounts from Internet communications services. You can see this in the slide Snowden leaked.

http://www.washingtonpost.com/wp-srv/special/politics/prism-...

https://medium.com/@alecmuffett/how-to-talk-about-prism-and-...