The stains may be etched into the glass, and thats why you cant remove them. It may be best to replace them.

It is probably etched(glass cancer). I think it may be permanent or very costly to restore.

If it's hard water stains from lime, it can be corrosive. I agree that they may be etched in.

Agree it could be permanent. You might also try equal parts water/vinegar. Spray on, wait 30-40 minutes and rinse. WD-40 could be another option.

I too think it is permanent etching and to avoid that when you get new make sure the shower glass is squeegeed every time the shower is used and clean well once a week.with a vinegar solution

You could add window film on the outside of the doors to hide the spots.

But I have had the best luck with this stuff. My sprinklers sometimes hit the outside of my windows leaving spots and my window cleaner also uses this to remove them. There are other products with the same name, be sure it's this one. Non toxic, easy on hands/skin, smells great and really effective.

Link below:

Bio-clean

If it is buildup you should be able to find some chemical that can remove it. If it is etched into the glass you would have to replace the door. If you do replace it will suggest that you use a frosted glass to help minimize the impact of the hard water.