Oops, we're not supposed to clean the rimfire barrels? I've been cleaning mine like a normal centerfire rifle after every range trip.
Generally speaking, a .22 bullet leaves a bit of wax in the barrel after each shot, and each subsequent bullet scrapes a bit out and leaves a bit of its own. Thus the condition of the bore stays essentially the same for hundreds if not thousands of shots.
So at least with lead bullets, it's rarely necessary to clean the bore. Many shooters don't do it at all. Others after every few thousand rounds, once a year, or some other arbitrary schedule. I would suggest bench testing the gun every 500-1000 rounds. If accuracy falls off at some point, cleaning may be indicated.
If you pay close attention, you may notice that a gun might go a bit wild after being cleaned. .22s are somewhat notorious for needing a few -- or even a few dozen -- rounds before "settling down". This is because the condition of the bore is changing between each shot, until that waxy equilibrium is restored.
Actions, bolts, etc. do need to be cleaned more often, or they will begin to malfunction. I will generally do this every thousand rounds or so, depending upon action type and clean/dirty ammo.
I don't know how much if any of this applies to copper washed/plated .22, as I have never seen any reason to use it.
HTH!