It is a distributed system. You still have to have shared remotes in order to integrate branches.
You're also not limited to one shared repo by any means, so even synchronization between multiple developers can continue when github is down. To avoid problems it's of course a good idea when one of the repos is a "master", and if using a secondary repo for synchronization when the master is down, you'll probably want to switch to unique temporary branches for the purpose.
(I'm not affiliated Bitbucket in any way. I don't profit from them doing well, aside from them sticking around to keep providing free hosting for me.)