BuddyPress.org

The short answer is that you take the code that creates the activity post box (located in buddypress/bp-themes/bp-default/activity/post-form.php) and put it somewhere into one of the members/single profile templates, possibly home.php. I’ve done this on http://commons.gc.cuny.edu (though you won’t be able to see the post box without a login). It took quite a bit of playing around to get it to work right, especially to get the necessary Javascript in place.

I’m hoping @apeatling will post a quick tutorial on how he did it here on bp.org. I’m planning on writing something up about how I did it, but I’m sure his method was cleaner.