Hi, I am most probably not someone who would be looking to do this, but as advice I think it would be helpful to others if you can outline a few things in a bit more detail. Such as –
1. What genre is your writing?
2. How would you describe your writing style?
3. What does ‘unfinished’ mean? Finished but not good enough? Stops halfway as as plot ideas dry up? Not even started just at the concept stage? etc
Then once you get further into it you need to clarify the rules of engagement with your writing partner. I did this some years ago, working on script writing with a very good friend. What helped was that we already knew each other very well.
Our way of working was to map out the plot, then slice up the action into scenes (Chapters) and agree which one of us would first draft each scene. Then when we re-grouped we swapped output so each of us could edit the work of the other. Our ground rule at this stage was that we never argued over something that that other wanted to cut. It just got deleted and we moved on.
Good luck with finding a partner…or partners.