The institution where I work has been considering the Turnitin products lately. It has been an interesting process analyzing what it can and cannot do, and how to avoid a confrontational situation like that described in the article. First of all, Turnitin can't detect plagiarism. It is a text matching software suite, and can detect commonality between works. Plagiarism is a social phenomenon, and can't be dealt with by an automated tool.
Where the product really shines in my opinion is when it is used as a teaching tool. Students are permitted to submit their assignments to Turnitin before they submit to their instructor, and they get back their originality report from Turnitin. Then they have the ability to *learn* proper attribution and citation with the help of this tool. When a paragraph gets a low originality rating, they can look and verify whether they have correctly cited their source material, and if they have, then they're good to go. When used like this, Turnitin becomes a valuable teaching tool that is appreciated by the students rather than something they try to fight against. And what is the goal in the end? It's not to throw students out of college, it's to make sure they understand how to attribute things correctly, and make them better writers.
To answer the question about what happens if there is a match to a paper from another institution: In that case, an instructor account can request a copy from the other paper's instructor via Turnitin. If the request is granted, the appropriate sections are forwarded to the requesting instructor.