I can give you Gizmo's answer (my male Eclectus), he will do loops, flying to the bathroom and then back to me if he hasn't had a shower and he wants one.
For Gizmo it is a minimum of 3 times a week, and then during moulting its once a day. When he is moulting he will go right under the water, wings drooped until he is soaked through, when he isn't moulting he does like to get fully soaked, but his priority is to run around having fun.
For those struggling with getting their birds to accept the shower, my suggestion would be introduce them gradually to the shower head (no water), my two are fascinated by the reflection. Once they get interested and start deciding "hey I wonder if I can chew this, start introducing them to it with the shower dripping. Once they start playing with the drop, or drinking it, you can increase the volume of water slowly. Obviously as the parront you will know best how quickly to proceed through the steps.
Depending on your birds feathers I don't think a sprayer is going to get them wet enough to the point that Gizmo seems to want during moulting, Pebbles can be drenched through in about 30 second under the shower, but Gizmo beads water for some time before it gets through (that would be a lot of water through a sprayer).
So the advice I would give, is try to shoot for daily if he is moulting, and during this period try to get him as wet as possible, if Gizmo is prioritising getting soaked through over running around like a fool, it must be important

.
Lastly (and thus should go without saying), I am careful not to leave my birds wet on a cold day. They love the hairdryer, and I stopped using it when some suggested it might dry the skin out, but during winter I still use it, as I worry more about them getting a chill. I haven't noticed any skin issues with using the dryer during winter, but the area I live in tends to be humid, so perhaps that helps.