In this house we both shop and we both cook. I probably do about 70% of both, because I have more time. We rarely write lists. He'll ask me if there's anything we need sometimes, but I generally am happy for him to buy what he fancies. If I can't think of anything to cook out of the random selection in the fridge, then he cooks (he is far more inventive than me!).
Occasionally we end up doubling up with things we've both bought, but it's not really a problem. DP does a physical job, and he's like a human dustbin, so not a lot goes to waste!