The degradation is intentional. In Windows phone, speech to text and quiet hours doesn't work, and the accuracy of shape writing is significantly reduced, when Cortana is switched off.
Ms is trying to turn Cortana into the Google play services of Windows So they can consensually(forcefully) collect data from every windows device.
I'd guess that means speech to text and shape writing run at least partially on servers? Not hugely surprising but it seems really bizarre that quiet hours would need it. Probably some integration with cortanas contact database so that some people's messages come through? Even that seems incredibly lazy and ill thought out.
Some years back Google went through a period of trying to hitch every new, and a bunch of existing, services to Google Plus.
It just so happens that the guy in charge of G+ was a former MS exec. And over at MS it was typical to attempt to hitch your project to any other project coming down the grapevine (the Google people actually called this "cookie licking" on stage after he had left the company).
And if you look back you could see this going on with .NET, you can see some semblance of this with Surface, and now with Cortana.
I guess it can be seen as some kind of in-office bandwagon politics.
There's no need to guess, at least in the PC version of Windows 10 Microsoft explicitly says this is the case. No one should be surprised that this is the case for Windows, given that other speech/shape recognition is similarly degraded with cloud support turned off.
Ms is trying to turn Cortana into the Google play services of Windows So they can consensually(forcefully) collect data from every windows device.