It's nice to see more people who are into co-writing on here. I enjoy co-writing, as long it's with someone who is equally committed and on the same wavelength as me, writing and idea-wise. It's not always easy to find the perfect collab partner.
As far as co-writing style, I like to chat with my co-writers on IM to make a plan. We brainstorm ideas and put together an outline for how they go together. I'm not a fan of just winging it, waiting to see what the other person comes up with and then picking up where they left off. There's certainly a challenge in that, and the spontaneity can be fun, but in my experience, it leads to more writer's block, plot holes, and a story full of twists and turns that don't always make the most sense. With a serious story, I like to plan it out in advance and make sure we're both on the same page as to where it's going.
I've written different collaborations different ways, but usually, we're each responsible for certain chapters or parts - I've never done it where we alternate paragraphs; I find it flows better if one person at least writes a whole scene. Some stories we've just alternated chapters, and others, we're each responsible for certain characters and write the scenes about those people. With my big collaborations, like 00Carter and Code Blue, we outline all the scenes in each "episode" together, and then everyone goes back and claims which scenes they want to write.