I remember hearing about the pirated movie thing but from what I remembered it was dropbox. This is the story I'm thinking about.
Dropbox isn't snooping, but it knows when you try to share copyrighted files. I don't remember any instances of that happening with OneDrive. And the website checking I think is also something old, I believe IE has done that for years, it checks them against known security risk sites. At least that's my understanding. I feel like a lot of this is old stuff but it's being presented in a new and more upfront way so people are getting all worked up over it even though it's probably less hidden now than it used to be.