My wife and I applied for GEORGIA concealed weapons permits the day after we got our Georgia driver's license. (aside - the DMV here in GA is head and shoulders above the DMV in CT; the process was clearly defined and executed with courtesy). The processing for our weapons permit took place at our county Admin building. We simply walked in, found the correct office and applied. It's handled by the county probate court. Presented ID including proof of residency, (our new drivers licenses), then fingerprints, picture, form filled out, check handed over - and she said she would call us in a week to ten days. Eight days later she called and told us that we could come pick up our permits.
My only basis for comparison is my CT permit (which has been converted to a non-resident permit). When I got my CT resident permit, it was through my local police department. Some of the people I dealt with there were awesome, a few weren't. There were specific days and times that I could apply, and it took 8 weeks to process.
From my perspective, it's the same process in each state; I don't see any valid reason for CT taking 8 weeks as opposed to GA's 8 days.
EDITED to add:
The CT permit I got from my local town was valid statewide, but only good for 60 days. I had to go to a state office to exchange it and more cash for my state issued permit. CT cost me $50 plus $19.25 on the original app plus $70 to exchange my town issued permit for a state issued permit.
GA issued me a statewide permit for $77.50.