I feel like you become a songwriter when you claim that it's sort of like a switch flipped, and you're always writing. Even in your sleep, you're always thinking about it in the back of your mind. The true writing - when you're officially writing - that's just when its front of mind, but its always there. You're always listening for a hook.
I believe the better I can write on my own, the better I can be of service in a co-write. The big thing with a co-write is trust, and it's not so much what you get the first time you sit down with that writer. It's the relationship that you build that you're in. So you can trust throwing out all your ideas and how strong you think they might be.