The answer to your question regarding agency sharing of files is going to depend on the level of coordination amongst background investigators at the various agencies. It would seem at least among local law enforcement agencies, that the greater the physical distance separating them, the less likely such coordination will be.

With regard to requesting your applicant file (assuming the records are not with a federal agency) your rights to such records will be governed by state and local laws. For example, in my state (California), the pertinent law is the California Public Records Act. If you've applied with a federal agency, you may request your records under the Privacy Act and the Freedom of Information Act.