I mean you’ll notice in this image it specifically shows you talking to your phone not to the Google home:
Your phone is able to process voice, figure out that there is a local fulfillment path for your command and then send fulfillment request over to the Google home device (which then runs your nodejs app to make API calls to HA). It specifically doesn’t show you talking to your mini/home.
I don’t have a source because like I said, Google is being kind of opaque about this. But see here look at this phrasing at the top of your link:
The Local Home SDK enhances your smart home integration with Google Assistant by adding a local fulfillment path to route smart home intents.
“Route smart home intents”. The Google home devices aren’t doing nlp locally, they are used for routing already processed intents to somewhere locally. So if a device on your network is creating intents and sending them (like say the phone in that picture or some other smart device) then sure it stays local. But if you talk to your mini (like most of us do) then currently it needs help figuring out what you said.
But I do remember them announcing new chips that were supposed to be capable of doing limited nlp relatively recently (maybe last I/o?). So the newest devices might be capable of this? Else maybe it’s coming soon™?