The service can verify the user's cert against the identity provider's public key with this information. The service initially contacts the identity provider to get the provider's public key, so the identity provider can infer that someone is attempting to log into the service as one of its users.
Once the service has the provider's public key, it can be cached by the service until it expires, and while it's cached, the service can verify logins without contacting the server again by checking that the cert presented by the user to the service has a legitimate signature from the identity provider's public key.
So if someone logs into HN using their Gmail ID (assuming both services grow support for the protocol), all Gmail knows is that at least one person attempted to log into HN using a gmail address. Who was trying to log in, whether that login was successful, and the number of other people who logged into HN with Gmail after that, are all facts that would be completely hidden from Google.