Are you referring to the so called “admin bar”? Black bar with “Howdy, username” or similar on the right? The only other input field I see is the search box. FYI, the admin bar is not unique to P2, all WP installations have it available to registered users.
The key phrase is “available to registered users”. Much of the content is configured for a specific user’s capabilities, so it’s not possible to display it for non-registered users. About the best you can do is add custom content to the theme’s header.php file by creating a child theme. The content can be styled to look exactly like the admin bar. Control its output with !is_user_logged_in() so that registered users will still get the real admin bar but not the replica version.
Do you realize that allowing anyone to post on your site will make it a popular target for spammers? You’ll want to at least install safeguards much like we do for comments to keep the spam from getting out of hand. A single automated spam-bot can post staggering amounts of spam in a very short period if not somehow kept in check. There are thousands of such bots trolling for open sites like you envision.
