We just experienced a case where our helpdesk and the helpdesk of a partner company got into an autoreply war ( i.e. we forwarded a ticket to them, their helpdesk responded by creating a new ticket and sending a new ticket notification, we responded to it by creating a new ticket and sending a new ticket notification and so on ).
We believe that in the 'Edit Contact' page there should be an option saying "Do not send a New Ticket Created Requester Notification". This can be used, as a last resort, when you get involved in such a case.