I went and spoke to the professor about this, asking if I could email her the paper, rather than using this service. She said that if it was up to her, I could but it wasn't and I would have to talk to the head of the department. I went and spoke to the head of the department and she accused me of wanting to cheat, due to my objections to the policy. I pointed out that I objected to the service keeping the rights to my paper to use for checking against anyone else's paper. She all but said that she didn't care and that my choices were to use the service or to fail the class -- it was too late in the semester to drop the class without approval from the dean.
I ended up using the service, I was a freshman and didn't want to fail the class. Especially not so late in the semester. At the top of each paper was a notice saying that the work was copyrighted to myself and that I didn't give TurnItIn permission to use it. I doubt the service noticed, it is automated, after all.
I've since made a point to ask every professor in a class where I would have to write a paper if the department uses such a service. They've all said basically the same thing: "We don't use that service, don't worry," and one even mentioned the concerns I had as a reason why they didn't want to use it either.