Badger,
FAA Form 8310-3 definitely gets filed with the FSDO; it's the Application for a Repair Station Certificate and/or Rating.
You're referring to FAA For 8410-3. Use of this form is not mandatory, and the Air Transportation Operations Inspectors Handbook specifically encourages POI's to direct their charge certificate holders to come up with their own tailored forms to match their training program. (Very few operators do this, however).
Inspectors are not required to keep copies of this form. The company retains one copy, the airman gets the other. Typically the company keeps the origional. The FAA keeps their own records using the Program Tracking & Reporting System (PTRS).
An inspector may elect to require an operator to supply him or her with the forms in his capacity as POI. However, generally the forms are retained at the company level, and reviewed only during a base inspection, or audit.