Yes, this is correct behavior, when someone click “login” in that red box s/he will be redirected to login page and when login, s/he will be redirected to message page.
You can even add
add_filter( 'fep_using_auth_redirect', '__return_true' ); in your theme’s (child theme’s if any) functions.php so that you will be redirected to login page automatically if not logged in. So no need to click login.
You can also set who can access message system in Dashboard > Front End PM PRO > Settings > Security