It's SOP to advise the officer that the party being run thru the database has a concealed firearms permit down here in SW Utah as well. The CFP info shows up whenever they run your driver's license. Down here, they'll say "valid concealed firearms permit, expiring ####" or sometimes they'll say "valid 10-88."
Depending on why you are interacting with the officer, he or she learning that you have a CFP might make them relax a little, or of course it could make them more concerned.
The only problem I have is that sometimes the officer will ask for the firearm, and will run the serial #, claiming they want to see if it's stolen or not. By doing so, this becomes a de facto state registration of firearms, because now your name in conjunction with that exact firearm gets saved in the database.