When a guest (=not registered) user chooses from the menu one of the menu items, gets redirected to the login page!
Can't a guest user see the products?
Yes they should be able to see products
Check product configuration to ensure that products are "available to all"
Yes they are (Products->Product Information->Shopper group: available for all).
Check the Access settings of menu item. It must be set as Public.
Yes, the problem was in the menus' access level, I thought they all were 'Public', but they were not.
Thank you all!