My understanding of Web Speech API's implementation, especially the SpeechRecognition interface, is that it relies on each browser's vendor proprietary services - so possibly Chrome's "calls home" to the same infrastructure that power's other Google's voice-powered services, Edge's calls Microsoft's, and so forth.
I have confirmed this at least for Chrome, blocking all connections to google domains, and seeing SpeechRecognition stop working.
This led me to two questions to which I couldn't find answers anywhere:
Is any documentation on how browsers call home for certain APIs? (perhaps not limited to Web Speech)
Is there a quota for calls to Web Speech services? Both Google and Microsoft sell speech recognition and TTS as a service, so I would be surprised if there was virtually no limit to Web Speech usage as long as it goes through native browser's implementation calls.