What’s the right way to share data between widgets?

Presumably, these are widgets that you are writing. If so, basic PHP class methods should do it– set a static class property to hold your “overlap” data.

class Foo extends WP_Widget {
  static $overlap;

  /*constructs etc*/
  function __construct($id = 'twidg', $descr="Test Widget", $opts = array()) {
    $widget_opts = array();
    parent::__construct($id,$descr,$widget_opts);
    /*do stuff*/
  }

  function widget() {
    // set static::$overlap
    static::$overlap .= 'something-';
    echo 'test widget :: '.static::$overlap;
  }
}

You could even create a “parent” widget to handle the “overlap” logic for all of your “child” widgets. For that see:

• https://wordpress.stackexchange.com/a/101443/21376
• https://wordpress.stackexchange.com/a/100721/21376