Can ick be transferred through fish nets?

In theory ick can't be transferred after it has dried out or a couple days have passed, but there are other opportunistic organisms that can be on an old dried out net, and they tend to show up just when fish have been weakened by ick.

Dipping a net is boiling water sterilizes it and also softens the net material making it a bit easier to capture fish in. Jungle Labs sells a net sterilizer you can store your nets in. It's more convenient than boiling water. Either way be sure you are careful and keep any hot or chemical solutions away from children and in places where they are not likely to spill.

Isolating tanks and equipment for each tank is always good practice. When you can't, take precautions.

Sterilize it. You might get away with storing it in a concentrated salt solution but the safest way would be to have two nets.