Hi, I'm Beda, I had a (several) looks at this issue when Christian escalated it, and the main issue I had was that the Duplicate wasn't allowing me to deploy the site at first, but later also I found, the content of the Duplicate is significantly different from what I saw on your test site.
For example, on the Duplicate, I see the culprit View has an Ordering by a Custom Field that does not exist for the Post Type in question and hence not even admins sees content in the page.
Another issue was also some HTML condition, that was wrongly formatted.
But on the local site Toolset Access had no effect whatsoever over this, and also it can't, because this is actually a Feature often requested that Toolset Access misses: to control the views loop output.
On the testing site, however, as soon Access is deactivated I see content in the Spanish view as that test user.
When you take that View and insert it into a new page, and translate that page, the same issue happens.
This is truly strange, as Access, has knowingly no rule over Views!
And as soon I give Read rights tot he Series Post type for the role (one of) this user has... I can see the output in the front end.
And this all does not happen in the native language.
So the steps to replicate this issue should be as follows:
1. Create a user with maybe many custom roles
2. Install WPML and translate some content
3. Create a View of that content and show it as well translated
4. Visit as a user with no read rights over that post type, in a secondary language.
The user is expected to see content but if the bug is replicable, the user won't see
Before I dig into this, can I ask - on this test site - might I remove other plugins and themes as well?
I want to be sure I don't miss an obvious conflict first.
NOTE:
I have left the read rights for the training role ON, for that post type, Series so you can see it working now.