How can I create an RSS feed that includes “Private” posts?

I assume your theme already has a ‘feed-rss.php’ (if not, here’s how they work: http://codex.wordpress.org/Customizing_Feeds)

You’ll need to pass a few parameters to the query before you loop through them.

//Get all posts that are private or public
$the_query = new WP_Query( array( 'post_status' => array( 'private', 'publish' ) );
while ( $the_query->have_posts() ) {
    //do stuff
}

For more parameters: http://codex.wordpress.org/Class_Reference/WP_Query#Status_Parameters