When you publish or update a post containing a gallery shortcode, the post id is written to the gallery's XML file (in the 'wp-content/uploads/juicebox/' directory). The gallery will then be associated with that post and will display the images attached to it.
If you have the same gallery shortcode in multiple posts, then the post id which is written to the gallery's XML file will be that of the post which was most recently published or updated which contained its shortcode.
(Each gallery should be associated with only one post and you should not have the same gallery shortcode present in multiple posts.)
If WPML duplicates an original post to make a new translated copy, then the gallery shortcode will be present in two separate posts and I suspect that the post id associated with the gallery will be that of the new post. Therefore, when you try to view either post (the original or the new post), the gallery will display the images attached to the new post.
The fact that no images display suggests that there are no images actually attached to the new post.
To check, try editing the new post, click the 'Add Media' button and select 'Uploaded to this post' to see what images (if any) are attached to it.
It might still help if I could see your broken gallery live on your web server but I suspect that from what you are saying (and my notes above) that WP-Juicebox may not be compatible with WPML and would need to be rewritten to change the way it works.
Bear in mind that you could still embed a Juicebox gallery manually in a WordPress post using the baseUrl method documented here. Such a gallery should work fine with WPML.
I will certainly keep an eye on the thread you created in the WPML forum, though.