Where does the Media Library live in the database?

SELECT    
    p.ID,
    p.post_title AS title, 
    p.post_name AS post_name,
   (SELECT meta_value from wp_postmeta where post_id = m.meta_value and meta_key='_wp_attachment_metadata') AS meta_value
FROM 
    wp_posts p, 
    wp_postmeta m
WHERE
        p.post_type = 'post'
    AND p.post_status = 'publish'
    AND p.id = m.post_id
    AND m.meta_key = '_thumbnail_id'
ORDER BY 
    p.post_date DESC
LIMIT 5;

And then use PHP unserialize 'meta_value'

Select * from wp_posts where post_type = 'attachment';

Will return all the entries in the Media Library.
After the execution, you can export the result table as SQL, or CSV, or any other portable data format you like. Remember, if you are not sure if the entries already exist in your database, use the INSERT IGNORE statement instead of INSERT. (This is possible through exporting pan in phpMyAdmin or other MySQL clients).
Also, there are entries referring to the Media Library in each post, such as attachment images or thumbnail images, which are stored in the wp_postmeta table. Wordpress stores them so the media "attaches" to posts or pages. If you want those to be exported too, you will need to use something like this :

 SELECT * FROM  `wp_postmeta` 
 WHERE meta_key IN (
   '_wp_attached_file', 
   '_wp_attachment_backup_sizes',  
   '_wp_attachment_metadata',  
   '_thumbnail_id'
 )

And then you can export them to wherever you want. It is all I know about media library stuff in Wordpress.

The Media Library lives in both wp_posts and wp_postmeta.

• wp_postmeta contains the image URL
• wp_posts contains an entry for each image insertion into a post, along with the post ID.

Exporting and importing these 2 tables as SQL did not work for me - I received 'duplicate entry for key 7'...

Exporting and importing these 2 tables as CSV did work, using "CSV using load data".

Before importing, I emptied the 2 tables in the recipient database.