Either/or. They have a private cloud option that works faster and better than when its running on the low power chip on their "green" hub, but it does work. This costs some amount. It will work fine on just the local hub, but slower and more limited.
You can instead opt to install home assistant on a beefer mini pc. Something with a n150 chip or better is recommened. With it installed on that, the voice hardware apparebtly runs very well just locally.
I dont have one, I've just been looking into it recently, so take the above with a grain of salt.