One of the most obvious forms of collaboration is the picture book, where one person writes the text and another illustrates it, although the usual situation is that the two don't meet, and the illustrator takes the text and interprets/illustrates with their own ideas. The editor is the go-between. There are famous collaborations in fiction - Nicci French and P.J. Tracy, for example (NF is a husband/wife team and PJT is a mother/daughter team) - where two people work together to write a novel. It's an interesting question. Who does what? Do they take turns writing chapters? Does one create the plot and the other write the words?
My writing group has been working on a novella together. It started as maybe a long short story, is now sitting on around 33,000 words and will probably end up about 50,000 words, the rate we're going. Or more. We originally thought we'd do it as a different way to create a group anthology (the Victorian FAW has a yearly award for writing group anthology), but the characters and story have kind of taken off, and we are having a load of fun.
First step was to come up with a situation that could involve a number of characters, and put them in conflict with each other. We settled on a family funeral (like weddings and Christmas, these occasions bring out the worst in some people). Each of us writes one of the characters, in first person, and there are some other characters who appear in the story too but don't have their own voices.
The best part is the plotting. We sit around the table and throw ideas around about what might happen next. What will X do? Why is Y behaving like that? What will happen when Z finds out the truth about ...? Having plotted out the next few scenes and decided from whose point of view each scene will be shown, we go away and write, then bring back our bits and read them out. Often our characters will throw in something new (that just came out of nowhere in the writing!) which makes everyone else say, "Wow, that's great, that takes it in another direction. Now, how about ..." And we start the next round of plotting.
Some characters are behaving badly and so the feedback might be, "You need to show why he's doing that", or "Your character has been observing - now she needs to act and stir up trouble". We've introduced an unexpected romance, and someone else who thought they were in love is about to realise they were wrong. Other characters are getting desperate, or looking for reconciliation. It's all inspiring and energising, and feeds into our other writing as well.
What will we do with it? Well, we are over the word limit for the award so we'll rethink that, but probably we'll make it into a book and print enough copies so we can all have one each and some for interested friends and family. We're not expecting it to be published commercially - it's going to be too short, for one thing. But mostly we're going to continue having lots of fun!