Your choice basically comes down to whether you want it to be private or not.
If you want it open to the world you can't go wrong with either.
However, if you want it to be private you should go with bitbucket. Even then you are limited to 2 users in a free private repo.
My guess is you want it to be private and free (since you're a student) so your only other option there is to host it yourself. This would be a great option if you can pull it off. Remember, you don't need to have it hosted on a server to share and merge, there are plenty of commands for sharing patch sets through email.
Kiln http://www.fogcreek.com/Kiln/StudentAndStartup.html is another option you can get for free since you are student. Run by Joel Spolsky's company I believe.