Dropbox snoops through your files

One possibility is that Dropbox hashes your videos/photos, and compare your hashes to hashes of known shared images obtained from previous prosecutions, and reports matches to police. That's not exactly "snooping through your files."

Source: I served on a jury on a case like this once. It was very unpleasant.

Of course they snoop on your files. You're uploading to their system and they really have no incentive, nor requirement, to give you privacy. Anyone who uploads sensitive files, from porn to bank statements, to Dropbox should know without a doubt that someone is probably going to see it. The only way around that is to encrypt locally, then upload.

Obviously I'm all for catching disgusting people like this but how is this possible without gross violation of your privacy?

Any user should know that illegal material stored in the cloud is not private.