Is there a way to show attachment IDs on the attachment info screen?

You could hook into 'attachment_fields_to_edit' and just add a row:

<?php # -*- coding: utf-8 -*-
/**
 * Plugin Name: T5 Show Attachment ID
 * Version:     2012.06.04
 * Author:      Thomas Scholz <[email protected]>
 * Author URI:  http://toscho.de
 * License:     MIT
 * License URI: http://www.opensource.org/licenses/mit-license.php
 */

if ( ! function_exists( 't5_show_attachment_id' ) )
{
    add_filter( 'attachment_fields_to_edit', 't5_show_attachment_id', 10, 2 );

    function t5_show_attachment_id( $form_fields, $post )
    {
        $form_fields['t5_id'] = array (
                'label'      => 'ID',
                'input'      => 'html',
                'html'       => "<strong>$post->ID</strong>",
        );
        return $form_fields;
    }
}

Result: