Match two taxonomies to display a specific content

You’ll want to have a look at get_the_terms().

<?php
  // You need to define the post id from the post you are outputting in the right bar so that the left bar knows how to match up the terms
  $country = get_the_terms( $post->id, 'countries' )

  $args = array(
      'post_type'     => 'side_menu',
      'posts_per_page'=> 1,
      'tax_query'     => array(
        // 'relation'    => 'AND', /* I don't think you need this unless it has another purpose */
        array(
          'taxonomy' => 'countries',
          // This only works since you are selecting just one term per post
          // Works by retrieving the first and only term id from the post in the right bar
          'terms'    => array_shift(array_keys($country))
          )
        )
  );
  $the_menu = new WP_Query( $args );
  if ( have_posts() ) : while ( $the_menu->have_posts() ) : $the_menu->the_post(); ?>
?>