There are no privacy rules specifically for landlords about applications, but see the guide below that has information about how they have to treat your credit information. Generally. while the right of privacy is waived for the purpose of the application, it is not waived as to the world, and privacy laws do protect tenants from a landlord's sharing of that information with others.
If your landlord tries to show you someone else's rental application, just say you're busy and by the way, not comfortable being shown other people applications because they have privacy rights. Maybe he'll get the message, And never tell your landlord anything personal, since he know he's an indiscreet gossip.
Disclaimer: Please note that this answer does not constitute legal advice, and should not be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue. This answer does not create an attorney-client relationship.