How to communicate with a third party from a Freshdesk ticket without involving the customer?

If you are looking to initiating a private conversation with a third party vendor who isn't a part of your Freshdesk Account, you could use the Forward option on the Ticket details page. 

Please navigate to the Tickets tab -> click on the ticket on which you would want to perform this action -> and click on "forward" next to the reply option.  

This sends the entire thread or individual reply (depending on which Forward is used) to the third party and it would not be visible to the customer. A reply to this email will be added as a private note to the ticket, which would also not be visible to the customer.