Absolutely. If I own my data, someone processing this data on my behalf has no right or obligation to scan it for illegal content. The fact that this data sometimes sits on hard drives owned by another party just isn't a relevant factor. Presumably I still own my car when it sits in the garage at the shop. They have no right or obligation to rummage around looking for evidence of a crime. I don't see abstract data as any different.

Our major privacy blunder was accepting scanning of private data in any context. The fight should be for the absolute privacy of personal data. Where the scanning happens is mostly irrelevant.