Hiya! While I have seen some people where you could see the end of the screw, the majority, no, you can't see it. It sounds like that particular screw is a bit too long for your nose
Sounds like you got some petite nostrils
I've had a nose piercing with a screw, and the end was hidden up inside my schnoz heh! (I've also had several other piercings too, but now I look more like Modern Preppy than Modern Primitive, ha!)
I'd have 'em swap it out for a shorter screw, or one they can easily bend & adjust to fit you. Especially if it is driving you nutty. It shouldn't be tickling you. That would drive me crazy too. Perhaps they can adjust the one you currently have?
Side note: If your jewelry has a Tiffany setting wrapping a stone, I just gotta say: watch out for catching it on the terrycloth loops of towels or washcloths! (Seriously! Oww! Speaking from painful experience!)
Good luck and healthy healing!