I wouldn't say this is laravel specific.
I would recommend that any sensitive data be stored in a session as it's stored on the server and not the user computer. The problem with cookies is they can't be relied on since the user can manipulate then and even delete.
Cookies to me are meant for things like remembering a user or a longer term storage for non sensitive data.
The way I normally do things is once a user has successfully authenticated store information I need about the user while they are logged in. In laravel 4 it's easy Session::put('key', 'value');