firebaser here
There is currently no native way to run your Cloud Functions on your own server. You can run the Firebase emulator suite to emulate functions, but those are not ready for production use.
The more common approach is to set up your own server, using something like Express. But you won't get the other trigger types that Cloud Functions offers, as those are only available within Google's infrastructure at the moment and you'll have to take care of scaling with the load yourself too.
