How do data protection laws apply to faculty applications?
I'm not a lawyer, but in the US at a federal level there is no legal expectation that job applications will be destroyed at the end of a job search. In fact it's the opposite, there are some laws (such as the Americans With Disabilities Act) which require employers to retain hiring records, which includes application materials, for a period of time in order ensure such records are available if someone were to accuse them of discrimination. 

Employers are free to retain such records for longer periods of time if they so choose. For example, it's very common for large corporations to retain job applications indefinitely so they have a permanent record in case you were to apply multiple times to the same place.

In practice, there is a general expectation that academic job searches are kept private, but probably not to the point where job applications are destroyed. My department is required to keep hiring records (application materials and any written materials we make about our applicants) and turn them over to our university HR department at the conclusion of our search. From there I assume the university may keep them indefinitely if they chose to do so. I don't think there's any law that makes it illegal for the department to disclose their potential list of applicants (or even their application materials), but doing so would be violating the social custom.

Academic departments tend to be very close-knit, with faculty serving together and depending on each other for many years. Announcing one's intention to leave a department is a sensitive issue that is seen as that person's prerogative to disclose in their own way and time. More importantly, faculty should be free to look at other institutions and entertain the idea of leaving for a different department without committing to that action or disclosing to their current department that intention, which could have major impacts on the way their current department advances/promotes/treats them in the future. For this reason it's a major faux pas to discuss potential applicants outside of the department, but there's no law against it.