You can use Facebook Chat API to send private messages, here is an example in Ruby using xmpp4r_facebook gem:
sender_chat_id = "-#{sender_uid}@chat.facebook.com"
receiver_chat_id = "-#{receiver_uid}@chat.facebook.com"
message_body = "message body"
message_subject = "message subject"
jabber_message = Jabber::Message.new(receiver_chat_id, message_body)
jabber_message.subject = message_subject
client = Jabber::Client.new(Jabber::JID.new(sender_chat_id))
client.connect
client.auth_sasl(Jabber::SASL::XFacebookPlatform.new(client,
ENV.fetch('FACEBOOK_APP_ID'), facebook_auth.token,
ENV.fetch('FACEBOOK_APP_SECRET')), nil)
client.send(jabber_message)
client.close