During my very last tour in the Navy, I manned the base armory on duty days. I'd be lying if I said that I remembered the whole procedure (this was eight years ago) but there was additional paperwork involved beyond checking service weapons in and out, enough that we didn't like processing it. I seem to remember it not going anywhere - it was mostly to establish a chain of custody and so that we could keep track of personally owned weapons during inventories. (They got inventoried the same as the service weapons did.)
A requirement for storage was that the weapon show up with a lockable hard case. The person checking it in showed that it was unloaded, locked the case, and that was the last that we saw of it. We did not have keys to the cases. Other posts might do things differently. When we inventoried these weapons, we were basically checking that the case was there, it was still locked, and it had the appropriate tag on it.
The inconvenience came from our hours. The armory was not manned 24/7. It operated during normal business hours Monday through Friday and re-opened around 2230 for the swing shift/graveyard turnover, but personally owned weapons would not be checked in or out during that time as a matter of policy. Likewise, the duty armorer would occasionally come in on weekends (besides security force turnovers) to check in/out match or ceremonial weapons and was essentially on call, but that number was not common knowledge. Rarely someone was PCSing in on a weekend; in that case, security would give us a ring and we'd head on down to check it in. This was not done with things being checked out.
Yes, it's inconvenient, and yes, it's there for a reason. Trust me when I say that your base armorers would much rather not deal with your guns, as they already have plenty to keep track of.
Oh, and before I forget...that paperwork didn't go anywhere. It was strictly for internal use. I think that we sent copies to security but don't quote me on that. Our primary concern was making sure that the owner's gun was unloaded and and available if the owner came looking for it, and the paperwork was meant to facilitate that.
On the ship, things were...well, different. That's probably a topic for another thread.