Dave's remark on switching to the student's shoes and contemplating what you can get out of a co-supervisor is, in my opinion, a very useful one and should be of primary concern to students.
Let me put some personal perspective on this. I, as a senior post-doc, act as a co-supervisor of two PhD students (with some international twist to it within the EU context) and have colleagues who do the same, hence my observations below. Of course my view is one of a junior researcher, so take it with a grain of salt.
My benefits from being a co-supervisor are the following, in the order of subjective importance:
new horizons: the students act as drivers and catalysts of research topics I do not necessarily care for in a very personal and deep manner. It means that I can broaden my horizons and get somebody pushing me towards learning something new. That is a good thing for me as a curious person, as well as for my career.
publication record: given the first point, obviously, if the students work well and our collaboration works well too, since I get as much influence on their work as they allow me, or ask me to, we get together some useful stuff done and get papers published. I would stress the word "together", where I rather take the passenger's seat and try to help wherever necessary, but the crucial decisions are student's. The finished and delivered projects, as well as papers are obviously a good thing for both of us, too.
project leadership experience: often the collaboration is in a context of a project, where, as the more senior guy,I would take the role of a project manager, or a team leader. Of course this gives me plus points to the CV as well, not speaking about learning how to do this kind of work. Another good thing for my career.
soft skills: by doing the above and by that closely collaborating with people with whom my bond is tighter than just a joint interest (as it would be with a member of my community from a different institution), in a way, we are supposed to work out our ways along each other. At least for me, that is a good training too and good for my life and career, whatever twist should it take in the future.
should the collaborating partner become a friend of mine during the process, I would add it as a benefit too. But obviously this one is not everybody's piece of cake.
All in all, the points above boil down to a single one: being a co-supervisor means for me to become and act as a senior buddy to the student and prepares me for running my own lab/group, should it become reality one day.