I dont think so, When we log out the current session and cookies get deleted so that user can't view anything without logging again, but when we are logged out from the site we still can send message and read new message from clients untill we reload the messaging page, I think before sending a message or recieving a message the system should check if the user is logged in or not.
You see it but is it actually sent? The customer can confirm that.
I've suspected that part of the messaging system issue is the sessions, so people type a message and it never goes through because they're logged out. This would follow along with my theory, but I'm just guessing.
At 3:20 it prompts to remember the password. Maybe its something to do with how the browser saves cookies/password...
I could replicate the exact behaviour on facebook. Dont have someone to check this on upwork