Facebook messages from non friends
Facebook filters messages in two inboxes, depending on the sender of the message.
Messages from Facebook friends
Messages from Facebook friends appear in the Messages feed which can be accessed by the small Messages icon in the top right section of the toolbar. That sections gets highlighted when you get a message. You can also get an email or text message when you receive a message, depending on your settings related to notifications.
Messages from non Facebook friends
If someone who is not your Facebook friend sends you a message, than Facebook will put that message into a second inbox named Other. You wont get any notification, Facebook remains quiet.
On the website the Other inbox is found under the Messages notification tab, or within the Messages window.
You can also navigate directly to the Other inbox by going to: www.facebook.com/messages/other/
If you find an email you would like to interact with on mobile – you can tap on the Actions drop-down and click Move to Inbox.
Otherwise, if you don’t access this inbox and check for new messages, they will just remain there in the Other inbox.