When you install the Zotero Connector in your browser, your browser will tell you the permissions required by the extension:
This is the standard permission an extension needs to be able to interact with the content on webpages as you browse the web. The Zotero Connector needs it to detect content on the page and update the save icon with the detected item type (e.g., a book icon for “Save to Zotero (Amazon)”, a PDF icon for “Save to Zotero (PDF)”).
No data about the pages you visit is logged on your computer or sent to Zotero servers unless you click the save button, at which point the Connector will save data and files either to the Zotero program (if it's open) or your online library on zotero.org (if you've given it permission to do so and logged in).
If saving fails after you click the save button and “Report broken site translators to zotero.org” is enabled in the Zotero Connector preferences, the Zotero Connector will send an error report, including the URL, to Zotero servers.