Not owned one but looked into them before I got my last bike.
The older bikes are expensive to service, the newer ones not so much but still more expensive than your average Honda/Yamaha. I think every two years or so they need a belt service which requires the engine to be removed and is something like £600-800, though I think that's just the older bikes.
Generally servicing them carrys a premium over other bikes just because they're Ducatis from my previous research.
They have dry clutches which for me sort of ruins the bikes but I'm sure all the Ducatistos would claim it's part of the magic, they just sound like a bag of spanner at idle to me.
Which model were you looking at?