Herkimerhunter,
The paddlefish is protected in many waters, like here in Ohio. But I do know that out in Montana, where the world record comes from, you can shoot them with a bow. Last I heard you got a tag for one like a deer tag. Kentucky and Indiana allow you to shoot them, too I think.
Where I am from in Missouri you can't shoot them but spoonbill snagging is a deep-seated tradition and a hell of a lot of fun once you hook one of them. Just be careful because in Missouri you aren't allowed to posess the eggs at all. They must be discarded when you clean the fish. Back in the 70s and 80s guys were netting them just for the eggs as they make very desireable caviar. So the state passed the law that you can't posess the eggs at all, even if the fish was caught legally...which is too bad because those eggs make some of the best catfish bait there is. But it's not worth the hundreds of dollars in fines if you get caught with them.
They are truly marvels. My best is 71 lbs and I have been in the boat when a 96 pounder was taken.