Hello Sebastian,
In the case of the Nuntio Content module it is working, but I can only see it when I am logged out.
Let me explain.
I have placed the Nuntio Content module on a page of a 5.6 DNN install that uses core content localization.
I have entered content in the Nuntio Content module for each language tab I had (French and English).
Then I went in the page settings, opened the localization section and I clicked "Ready fo translation" which added a line for the Nuntio Content module in the "Module Localization" table. (see image below)
It is important for the "Is Detached" column to remain unchecked (red x).
I update these changes and go back to my page.
To see it working, I need to log out.
After logging out, I can switch culture with the language switcher and see the results.
The content of the Nuntio Content module gets dynamically localized.
If I try to switch cultures when I am logged in as a superuser, content of the Nuntio Content module remains in my default language.
I have the same "logged in, logged out" issue with the DNNtext skin object in DNN 5.6.
Localization doesn't seem to work when you are logged in as a superuser, but when you log out, everything works.