Loop through widgets in sidebar

The registered widgets in your sidebar-1 might be

[sidebar-1] => Array
    (
        [0] => categories-2
        [1] => archives-4
        [2] => recent-comments-4
        [3] => calendar-2
        [4] => search-2
        [5] => archives-2
        [6] => text-4
        [7] => recent-posts-2
        [8] => nav_menu-2
    )

If you got for example the calendar-2 widget, then from:

print_r( $GLOBALS['wp_registered_widgets']['calendar-2'] );

you will get something like:

[calendar-2] => Array
    (
        [name] => Dagatal
        [id] => calendar-2
        [callback] => Array
            (
                [0] => WP_Widget_Calendar Object
                    (
                        [id_base] => calendar
                        [name] => Calendar
                        [widget_options] => Array
                            (
                                [classname] => widget_calendar
                                Loop through widgets in sidebar => A calendar of your site’s Posts.
                            )

                        [control_options] => Array
                            (
                                [id_base] => calendar
                            )

                        [number] => 2
                        [id] => calendar-2
                        [updated] => 
                        [option_name] => widget_calendar
                    )

                [1] => display_callback
            )

        [params] => Array
            (
                [0] => Array
                    (
                        [number] => 2
                    )

            )

        [classname] => widget_calendar
        Loop through widgets in sidebar => A calendar of your site’s Posts.
    )

You can therefore try this modified version of dynamic_sidebar() to call a widget that you have placed in a given sidebar:

/**
 * Show a given widget based on it's id and it's sidebar index
 *
 * Example: wpse_show_widget( 'sidebar-1', 'calendar-2' ) 
 *
 * @param string $index. Index of the sidebar where the widget is placed in.
 * @param string $id. Id of the widget.
 * @return boolean. TRUE if the widget was found and called, else FALSE.
 */
function wpse_show_widget( $index, $id )
{
    global $wp_registered_widgets, $wp_registered_sidebars;
    $did_one = FALSE;

    // Check if $id is a registered widget
    if( ! isset( $wp_registered_widgets[$id] ) 
        || ! isset( $wp_registered_widgets[$id]['params'][0] ) ) 
    {
        return FALSE;
    }

    // Check if $index is a registered sidebar
    $sidebars_widgets = wp_get_sidebars_widgets();
    if ( empty( $wp_registered_sidebars[ $index ] ) 
        || empty( $sidebars_widgets[ $index ] ) 
        || ! is_array( $sidebars_widgets[ $index ] ) )
    {
        return FALSE;
    }

    // Construct $params
    $sidebar = $wp_registered_sidebars[$index];
    $params = array_merge(
                    array( array_merge( $sidebar, array('widget_id' => $id, 'widget_name' => $wp_registered_widgets[$id]['name']) ) ),
                    (array) $wp_registered_widgets[$id]['params']
              );

    // Substitute HTML id and class attributes into before_widget
    $classname_ = '';
    foreach ( (array) $wp_registered_widgets[$id]['classname'] as $cn )
    {
        if ( is_string($cn) )
            $classname_ .= '_' . $cn;
        elseif ( is_object($cn) )
            $classname_ .= '_' . get_class($cn);
    }
    $classname_ = ltrim($classname_, '_');
    $params[0]['before_widget'] = sprintf($params[0]['before_widget'], $id, $classname_);         
    $params = apply_filters( 'dynamic_sidebar_params', $params );

    // Run the callback
    $callback = $wp_registered_widgets[$id]['callback'];            
    if ( is_callable( $callback ) )
    {
         call_user_func_array( $callback, $params );
         $did_one = TRUE;
    }

    return $did_one;
}

where you call it like this:

wpse_show_widget( 'sidebar-1', 'calendar-2' );

Another way would be to try to use the the_widget() function, where you would have to know the widget’s class name.