Hello Jan and thanks for the feedback. As you already mentioned the captions are supported only for CDS images for now. That happens because CDS images are based on a custom field that automatically retrieves captions from the CDS API.
On the other hand, local images use the default image field of Drupal which does not support captions by default so we would have to implement it in a custom implementation. In general the process should include adding a separate field for the caption and also overriding the default image template to include the caption or create a custom field formatter.
It is on the plans to integrate it but to be honest I hope to have a module that implements it since it is a quite basic functionality.