DUBLIN—When chat service WhatsApp last summer did an about-face and started sharing users’ personal information with parent company Facebook Inc., Ireland’s top privacy cop jumped into action.

Helen Dixon, Ireland’s data-protection commissioner, activated a 10-person investigative unit she had created to probe multinational tech firms, and she pushed to coordinate multiple probes across Europe. She has since put both Facebook and WhatsApp through rounds of questioning at her agency’s new offices in Dublin.

The...