"Go" is a way of saying "say" when you're quoting. We use it a lot, especially when telling someone about a conversation we had. It's usually used very informally. Ex. Me telling my friend a story: Me: so the other day I saw Josh at the movies so I go up to him and I go "hey josh!" But then he just stares at me and goes "who are you?" Like, we've been going to school together for years and he doesn't recognize me?! So then I just go "oh nevermind. I thought you were someone else"

To 'do your bit' is to make the amount of work and effort expected of you, when you are part of a team or there are multiple people required to complete a task. The author here is saying that the French government think they have done all that is required of them, and are saying that it is others' turn to do the same.