Hmm, it appears that the Matter integration page is misleading
Matter products run locally and always allow local control, with device control done without the need for any internet connection or cloud services. From a technical perspective, you can use a Matter-compatible device with Home Assistant without connecting to a vendor-specific cloud
AFAICT this is entirely untrue based on the spec. The protocol will always work locally but devices appear free to ignore local control or check with a cloud to see if they should listen to the command. They can also prevent you from even joining a device to your matter network unless you use their app/site.