Aside from all the usual bike stuff, on a Ducati you want to see some decent maintenance records, and especially that the valves have been checked regularly somewhere near schedule, and that the cambelts have been replaced within the last two years.
You will also want to make sure it ahs all the keys -- there should be one red key (important!) and two black ones.
Other than that, they're pretty normal, and very durable and reliable. A friend on mine has about 50K on her M620, and it's been a reliable bike. Mine's older and a little different, but the same basic design and mechanicals, and I'm over 200K on it.
1993 Ducati M900 Monster "Patina" (232,000 miles, so far) -- 1995 Ducati M900 (wife's bike) -- 1972 Honda CB450 (daughter's bike) -- 1979 Vespa P200 (daughter's scoot) -- 1967 Alfa Romeo GT Jr. (1300cc) -- 1964 Vespa GS160 (160cc 2-stroke) -- 1962 Maicoletta scooter (275cc 2-stroke) -- 1960 Heinkel Tourist 103A1 scooter "Elroy" (175cc 4-stroke)