Narrative Science stores (for a short period of time) a version of data used to generate a narrative. For example, if you request a narrative for a line chart, Narrative Science stores a post-processed version of just the data contained in the line chart. We do not receive any data other than what is needed to generate a particular story.
The Narratives for Tableau extension operates as a web service. This means that the raw data used to create the visualization is being sent to Narrative Science over the Internet to create your narrative. If you have firewall rules set up that prevent Tableau Server from communicating with the Internet, you will not be able to use the extension.