@Rikko I’ll answer myself here. :) It might help someone else.
The problem was the version of Wiring Pi, which is a dependency to this module.
If you follow the instructions on this module’s page, you get version 2.50 of Wiring Pi. But that does not work with Raspberry Pi 4. You need 2.52.
Here you can find details:
http://wiringpi.com/wiringpi-updated-to-2-52-for-the-raspberry-pi-4b/
My DHT22 now works well with MMM-LocalTemperature. 👍