WordPress offers no way out of the box to grab your sidebar widget settings and use them somewhere else in your site. Here’s a function to allow you to do it:

/**
 * Find a given widget in a given sidebar and return its settings.
 * 
 * Example usage:
 * $options = [];
 * try {
 *    $options = get_sidebar_widget_options('sidebar-1', 'recent-comments');
 * } catch (Exception $e) {}
 *
 * @param $sidebar_id    The ID of the sidebar. Defined in your register_sidebar() call
 * @param $widget_type   Widget type specified in register_sidebar()
 * @return array         Saved options
 * @throws Exception     "Widget not found in sidebar" or "Widget has no saved options"
 */
function get_sidebar_widget_options($sidebar_id, $widget_type)
{
    // Grab the list of sidebars and their widgets
    $sidebars = wp_get_sidebars_widgets();
    // Just grab the widgets for our sidebar
    $widgets = $sidebars[$sidebar_id];

    // Get the ID of our widget in this sidebar
    $widget_id = 0;
    foreach ( $widgets as $widget_details )
    {
        // $widget_details is of the format $widget_type-$id - we just want the id part
        if ( preg_match("/^{$widget_type}\-(?P<id>\d+)$/", $widget_details, $matches) )
        {
            $widget_id = $matches['id'];
            break;
        }
    }

    // If we didn't find the given widget in the given sidebar, throw an error
    if ( !$widget_id )
        throw new Exception("Widget not found in sidebar");

    // Grab the options of each instance of our $widget_type from the DB
    $options = get_option('widget_' . $widget_type);

    // Ensure there are settings to return
    if ( !isset($options[$widget_id]) )
        throw new Exception("Widget has no saved options");

    // Grab the settings
    $widget_options = $options[$widget_id];

    return $widget_options;
}

Drop the function in your functions.php. Note that it will find the first occurrence of your given widget in the given sidebar and return its settings.