Voice rec was local until about ten years ago. You could even buy software dictation products that installed from CDROM and didn't need a network connection.
I shouldn't need a network connection to do anything voice on my own device. It should only reach out to the network for things it doesn't have local knowledge of.
There's a reason voice rec has only now finally stuck.
It can be unacceptable for your privacy and security threat model but for most it makes 0 difference wether the training continuously happens in the cloud or locally on the NPU inside your phone.
Saving all the clips as much as people love to fear monger it is literally the strategy that objectively leads to the highest quality implementation on the market. Googles Assistant by far understands language most naturally and we owe that to all the free vocal training data google diligently uses.
Sadly, nobody will ever know if this is true as the choice doesn't exist.
Most people seem to like the Google Assistant for more natural interactions and only tolerate siri for her limited uses she can do.
But voice rec is not local. The reason it asks to store your voice is because they need to send it to their backend for recognition.