The implementation of the Web Speech API usually involves the specific browser vendor calling out to their own, proprietary, cloud-based TTS APIs. I say "usually" because, for a time, Microsoft used their local Windows Speech API in Edge, but I believe they've stopped that and have largely deprecated Windows Speech for Azure Speech even at the OS level.
Just to be clear, are you really saying that speech with text to speech is server hosted and not on device for Windows?
You could do text to speech on a 1Mhz Apple //e using the 1 bit speaker back in the 80s (software automated mouth) and MacinTalk was built into the Mac in 1984. I know it’s built into both the Mac and iOS devices and run off line.
But I do see how cross platform browsers like Firefox would want a built in solution that doesn’t depend on the vendor.
If the application is still using the deprecated Microsoft Speech API (SAPI), it's being done locally, but that API hasn't received updates in like a decade and the output is considerably lower quality than what people expect to hear today.
Firefox on Windows is one such application that still uses SAPI. I don't know what uses does on other operating systems. Like, on Android, I imagine it uses whatever is the built-in OS TTS API, which likely goes through Google Cloud.
But anything that sounds at all natural, from any of the OS or browser vendors, is going through some cloud TTS API now.
