Oh, never mind. I found the problem, which may help others as well.
In the documentation (in the settings page and in other forum posts, the shortcode for showing the photos in a photo set is like this:
[flickr_set id="0000000000000000"]
This doesn’t seem to work for me. However, removing the quotes around the set ID does work, e.g.:
[flickr_set id=0000000000000000]
Hope this helps others!