It depends. It would be fine if you not keep your sunglasses in chlorine water for to long. Or it may damages your polarized sunglasses. So, to be safe, you keep your polarized sunglasses away from chlorine water. And once your sunglasses splashed with chlorine water, you shall rinse off your sunglasses lenses immediately with warm water to ave your polarized sunglasses.
No. Chlorine won't damage polarized sunglasses as long as you don't put your polarized sunglasses in chorine water for a long time. I don't think it is necessary to wear polarized sunglasses to the swimming pool. As you go swimming, you only need to wear swimming goggles when swimming. Anyway, you don't have to worry about your polarized sunglasses. They have no problems at all.