What would cause the outside of a vinyl shower curtain to be sticky?

It might be something you are using to clean your bathroom with. You might want to rinse the curtain and the tub and see if it stops. If it does, it might be the cleaner you use. I can't think of anything else that isn't aerosol based.

I would try washing the shower curtain in the washer with a few towels then hang back up to dry

If you wash a shower curtain and ad fabRic softner, it may cause a sticky feel.

I wash mine in the washing machine about once a month. I think it's the humidity, etc. that causes the sticky feeling. I hang it back up and let it air dry.

Run it through the washing machine with white vinegar