Yeah, I've been there, done that
We actually went through 2 of them. The first one mildewed after only about 6 weeks, so when I complained to the company, they offered to give me another for 1/2 off (it was just past the 30 day guarantee). I decided to take them up on it and thought maybe I just hadn't washed it often enough.
Sadly the second one mildewed pretty quickly too.
So here is what I learned and what I suggest:
First, it is unlikely you'll get rid of any existing mildew spots. They'll probably just get worse. But first, try washing with vinegar, and sunning for a day or two. If they don't disappear, they probably won't.
You CAN, though, cut off the effected area and rehem the curtain. Make a point of wringing out the curtain after every shower before hanging it to dry. Let it hang outside the tub instead of against the inside. This makes a huge difference because it dries much faster! Also, wash it every 2-3 weeks and rehang to dry. In between spray with an anti mildew spray. I made a homemade one with TTO.
I eventually gave up on them because it was so high maintenance and the mildew kept returning. I was so disappointed!!! I hope you have a different experience.
I eventually went with a nylon liner and a cotton outer curtain. They still have to be washed monthly, but are much easier to maintain (and still aren't gross like PVC!).