If the company paid for your training, they own the records. FSI and CAE both send the training records directly to the company rep (usually the Chief Pilot). They do usually provide you with some sort of verification (logbook sticker).

There is certainly no legal obligation under the FARs for them to give you the records nor do they have to reply to PRIA on an employers behalf.

Don't know about CAE, but FSI has an online portal where I have access to and can download copies of my training certificates.