How to check what is not working in my chime

I've got an electromechanical chime at my business premises. there's a pushbutton outside in the hall and the doorbell inside, powered by the house power supply.

I've already checked the pushbutton by shortcutting the wire in the pushbutton: it doesn't ring.

So I guess the culprit should be the doorbell itself.

I don't really know whether it's using the house power voltage directly or transform it to lower voltage? (I'm in France, it's 220V here)

At my disposal I've got a multimeter which I've never really used.

How do I check which element could be wrong? Where do I apply the red and black probes of the multimeter?