In HuffingtonPost's Facebook Page account, the Wall tab shows all posts.
But in my user Account, the wall tab does not show all post or tweets for anonymous.
It only shows for a logged user (Facebook).
How to show the wall that contains the posts of all users, even if they are not logged in?
My privacy settings are set for Everyone.
The HuffingtonPost's account has an additional "Fan Page" configured, whereas you are a regular user. Facebook makes a distinction and therefore has different access options.
See this Facebook FAQ link.
You can create one for yourself it seems, if you like via this create page.
Quoting from the FAQ:
If you are the official representative of an organization, business, celebrity, or band, you can create a Page to represent your organization on Facebook.
