It doesn’t appear that is a feature that can be enabled only on the AirPods. According to the support page it must be enabled on whatever device (iPhone, Mac, etc) they are connected to to work.
According to Apple all dictation is processed locally on device though they do have a transcript log sent to Apple with a random identifier assigned. This is different than the audio recordings you can opt in allowing in the Analytics & Improvement setting’s page. You can delete request history associated by going to Settings > Siri & Search > Siri & Dictation History and tapping Delete Siri & Dictation History.