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The difference comes down to control. A Data Fiduciary decides why and how personal data is processed. A Data Processor only acts on the fiduciary's instructions. The fiduciary carries the responsibility either way.
Can one company be both?
Yes. It is a fiduciary for its own purposes and a processor when handling another fiduciary's data under instruction.
Who is liable if a processor mishandles data?
The fiduciary carries primary responsibility and stays accountable for the processor's compliance.