If you're talking about a data recovery service, they're necessarily going to at least see the *names* of the files. They're trying to recover the files after all. As for opening the files, while they're technically not supposed to be opening files that have nothing to do with the repair...well...things happen.
If the hard drive is actually physically broken, you have better odds of them not seeing your porn stash. When you RMA a hard drive, it's usually just recycled and you get another one - the cost/benefit to the company of trying to repair a broken hard drive isn't worth the bother most of the time.