[blog_id]      => 1
   [site_id]      => 1
   [domain]       => foo-multi-site.com
   [path]         => /site-path/
   [registered]   => 2014-07-31 14:51:09
   [last_updated] => 2014-07-31 15:51:56
   [public]       => 1
   [archived]     => 0
   [mature]       => 0
   [spam]         => 0
   [deleted]      => 0
   [lang_id]      => 0
   [blogname]     => Site Name
   [siteurl]      => http://foo-multi-site.com/this-site
   [post_count]   =>

$current_blog = get_blog_details();
if ( $current_blog->path === '/privatesite/' ) {
    //display log in or call template for private site
}
else {
    //do what you need for default log in
}