Does the module send my customers' data to Zendesk automatically?

Only if you allow it, and only for logged-in customers. When the "Pass Customer Data" option is enabled, the module sends the signed-in customer's name and email to the chat so agents know who they are helping. Guests and logged-out visitors send no personal data. You can turn the option off entirely, in which case the store passes no customer details at all. The conversation itself is always handled by Zendesk under your own account and privacy terms.