If you are a company where your business is providing disk space for random files of a subscriber, you'll need to deal with law enforcement. But it's not likely that the business will be liable for the contents of those files unless they completely ignore any credible notification of illegalities. For the average Joe running a random piece of software that allows him to sublet his excess storage likely would have a lot more scrutiny and a lot more hassles proving those files aren't his.
I look at it like someone being caught transporting drugs. If they were shipped via UPS, Fedex, or USPS, the courier and company aren't going to be arrested. But if you or I got pulled over and drugs were found in a box that we were "just delivering for someone", we probably would be.