Depends on the state. Missouri by law in fact prohibits any information release regarding CCP by the issuing sheriff unless as part of a formal investigation/ court order.
MO used to collect & include CCP data in the MO Uniform Law Enforcement System (MULES) via department of revenue ( = DMV) but after the MO HP and the DOR gave the feds the lists of CCPs in breach of state law ( purportedly for SS fraud/mentally ill folks with a CCP and therefore likely a firearm), the state no longer actually knows who has a CCP-only the SD where you live- decentralized for privacy.
AFA how to act during a stop- be calm, keep your hands on the wheel , turn on a dome light before you stop, roll the window partly down, if you are carrying either tell the officer in neutral voice (I have a CCP and I am/am not carrying, it is on my left hip in a retention holster). Although in MO you are required to inform unless asked....
Do exactly as the LEO tells you.
edit: you are NOT required to inform unless asked