The project is open source. I have not used it. But it's using a documented algorithm (noted in the RFC listed on the open source project page), and the authenticator implementations support multiple accounts.
The actual process is straightforward. The one time code is, essentially, a pseudo random number generator. A random number generator is a formula that once given a seed, or starting number, continues to create a stream of random numbers. Given a seed, while the numbers may be random to each other, the sequence itself is deterministic. So, once you have your device and the server "in sync" then the random numbers that the device creates, each time you hit the "next number button", will be the same, random, numbers the server expects.
A secure one time password system is more sophisticated than a random number generator, but the concept is similar. There are also other details to help keep the device and server in sync.
So, there's no need for someone else to host the authentication, like, say OAuth. Instead you need to implement that algorithm that is compatible with the apps that Google provides for the mobile devices. That software is (should be) available on the open source project.
Depending on your sophistication, you should have all you need to implement the server side of this process give the OSS project and the RFC. I do not know if there is a specific implementation for your server software (PHP, Java, .NET, etc.)
But, specifically, you don't need an offsite service to handle this.
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