Initial thoughts are that the answer would depend on the wording of the contract, as that should contain the level of detail involved in the service specs and - either directly or implicitly out of this - data ownership.
I've seen contracts that drill down to microscopic detail when specifying exactly what actions a Processor will undertake for a Controller, but equally I've seen contracts that say (for example) 'Here's a list of our Customers: it's been cleaned against the DMA lists. Find out what they think about X please - we'll leave it to you how you do it.'
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