How do I notify the requester of a received private Facebook message?
You can create a trigger to send an email notification to the requester once a Facebook private message is received. The message will be sent to the requester if there is an email address associated with the user profile.
You will need to create a trigger using the condition Ticket: Channel is Facebook Private Message and the action Notifications: Email user (requester).
Unfortunately, it is not possible to add the auto reply directly within the Facebook private message using Zendesk Support. However, you can set up an auto reply within your Facebook page settings. You can find further instructions here: How can I set up an auto reply for Facebook tickets?