I wouldn't be so sure, as France and Russia have a long, special history of diplomatic ties, dating as far back as when both were among the most influential countries of the Old World.
You guys build warships for Putin's war machine which is murdering civilians in Ukraine, something the French knew they were going to empower someday with arms like these. You don't buy helicarriers for defense, you buy them to easily attack your neighbors. Ties between these two nations go back hundreds of years. I would absolutely not assume the FSB and DGSE/DGSI aren't talking. I would assume they are bosom buddies.
http://www.defenseindustrydaily.com/russia-to-order-french-m...
Countries with militaries that enable each other have tight intel relationships. This French/Russia rivalry is non-existant.
I think we are between a rock and a hard place here. I'd be willing to use a VPN in a non-cooperating country to evade state surveillance, but it is hard to make sure that you don't fall prey to non-state (or even maybe state) actors.
I'd rather have Putin spy on me, than my own governement, or its allies. He has far less influence over my life, and I doubt that french and russian intelligence are cooperating.