In edit issue there is checkbox "Private notes". Usually these notes are not meant to be sent to customer.

Helpdesk plugin has configuration option "Send note by default". When this is not checked, user needs to manually check send note checkbox.

Now it's still possible that support person might accidentally click both "Send note" and "Private notes" and the message will go to customer.
We would like that if "Private notes" is checked, email is never sent to customer.

This could be achieved with new Helpdesk configuration option "Never send private notes". If that was checked, email sending would be skipped when "Private notes" is checked. Regardless of "Send note" being checked.
Obviously there is possibilities to make this more elegant and ask confirmation from the user, but I would be satisfied with just skipping the email.