I'm sure this isn't the best way to do it, but I was in the same situation in my last apartment. I put a long curtain rod over the french doors and hung heavy denim weight curtains over both doors. (Velvet would be great too.) On the other side of the doors, my housemate did the same curtain thing and had large pieces of foam (2" thick I believe) between the curtain and the doors.

It's not a permanent solution but it helped quite a bit. Good luck!

Maybe you could arrange your furniture with a wardrobe against the partition or a book shelf filled with books.

I put a peice of egg crate foam on either side, hidden behind a curtain.

I agree with Helena and Princess Fiona. Use some egg crate foam and a bookcase. My bedroom is part of a double parlor. I tacked up egg crate foam on both sides of the doors. On the bedroom side I have bookcases in front of the foam. On the living room side there are curtains. It's really cut down on the noise and I couldn't be happier.

you're going to need layers, i'd forst suggest a heavy curtain and then the bare minimum of a decorative screen. you could go further by putting a bookshelf across the gap and filling it with books to help dull the noise.