Custom search to display results within same page

I think you need to modify the code to display the content as of now you are not displaying anything in side loop.

<?php
if ( isset( $_REQUEST[ 'find' ] ) ) {
// run search query
query_posts( array(
's' => $_REQUEST[ 'find' ],
             'post_type' => $_REQUEST[ 'download' ],
             'paged' => $paged
             )
          );

        // loop
        if ( have_posts() ) : while ( have_posts() ) : the_post();
            // 
?>
<a href="https://wordpress.stackexchange.com/questions/286524/<?php the_permalink();?>"><?php the_title();?> </a>
<?php 

        endwhile; endif;

        // return to original query
        wp_reset_query();
    }
?>       

Please check if this helps you