Hello PietroNicola,
It sounds like you're unable to use dictation on your Mac without internet access after updating to macOS Catalina. I can understand why it would be important to be able to dictate text to your Mac without being connected to the internet, and I'm happy to help you resolve this.
There are two different ways to use dictation on your Mac: Voice Control and keyboard dictation. Voice Control allows you to be able to dictate without an internet connection, while keyboard dictation requires you to be connected to the internet. To make sure you're using the correct one, and they aren't conflicting with each other, can you tell me which step you are taking to start dictation after its been enabled? If you're pressing a keyboard shortcut to start dictation, then it sounds like you're using keyboard dictation and would need to be connected to the internet. You can turn this off in Settings > Keyboard > Dictation.
With Voice Control, dictation is active as soon as it's enabled, and you should see an onscreen microphone representing the mic selected in Voice Control preferences as seen here: Use Voice Control on your Mac. It appears you've already enabled Voice Control, so you should already see that mic on your screen. If it says "Wake Up" under the microphone icon, you will need to say "wake up" or click on it before it will dictate your text.
If you're ever connected to the internet and would like to use keyboard dictation, here is some information about how to set it up: Dictate your messages and documents on Mac. Let me know if you have Voice Control enabled and Keyboard Dictation disabled and still aren't able to dictate text without internet access.
Thanks for using Apple Support Communities.
Take care!