A separate page would be the simpler approach. But here’s a way to do it with a JavaScript alert box (taken from this approach).
Add a trigger to a logout rule like this:
[variable]siteurl[/variable]?loggedout=true
and add this to your theme’s functions.php:
function plr_logout_alert()
{
if( 'true' == $_GET['loggedout'] )
{
echo '<script>alert("You have been logged out.");</script>';
}
}
add_action( 'init', 'plr_logout_alert' );
The downsides are
(1) the user will see mysite.com?loggedout=true in the address bar, and
(2) the page is not displayed until they dismiss the alert box.
There are more advanced ways of passing messages onto the next page – let me know if you’re interested and I’ll cook something up.