<?php
  $department_terms = wp_get_object_terms($post->ID, 'departments',  array('fields' => 'ids'));

  if(!empty($department_terms)){
    if(!is_wp_error( $department_terms )){
      $args = array(
        'post_type' => 'staff',
  	    'tax_query' => array(
          array(
           'taxonomy' => 'departments',
           'field' => 'id',
           'terms' => $department_terms
          )
  	    )
      );

      $department_posts = get_posts($args);
      if($department_posts){
        echo '<ul>';
        foreach ( $department_posts as $post  ) {
          setup_postdata($post);
          echo '<a href="'.get_permalink( $post->ID ).'">'.$post->post_title.'</a>';
        }
        echo '</ul>';
      }
    }
  }
?>