The extension reads the content of the page you are on only when you open it and only to fill in the campaign fields. It talks to PushEngage servers to load your sites, segments and campaign data and to send campaigns. When enabled, it may send scrubbed technical error reports to help diagnose issues. It does not track your browsing, does not run on pages you have not opened it on, and does not sell or share data.
Full policy: https://www.pushengage.com/privacy/