Dishwashers are usually much more water efficient than hand washing.
It seems our sink fits about 25 liters of water. Let's say half of that volume is taken by dishes, so 12.5 liters. You need a second sink for rinsing unless you run the tap (which would be worse), so that's another 12.5 liters.
From experience, you need to change the cleaning side's water at least once if you have as many dishes as a dishwasher would fit. So the total would be 37.5 liters.
Now I checked on Bosch's website for a standard 60 cm size dishwasher and the water usage ranges from 9.5 liters (eco program) to 18 liters (most intensive/quick program). So the dishwasher wins by some margin.
Personally I don't rinse most dishes before putting them in the dishwasher, mostly just remove any solid bits of food with a knife or similar. So I'm inclined to believe that in my usage the dishwasher is much more efficient.
Generally, when hand washing dishes, hot water is used. Typically ~5-10 times more water is required when hand washing, which results in more water needing to be heated (and is less energy efficient as a result).
Ah, is that so? I've not had a hand washing experience where hot water was used, I thought everyone used cold water. Also, I'm surprised at the efficiency of the dishwasher. Do people leave the faucet running while they do dishes? What I do and typically see is an initial rinse, then water is off while scrubbing, then a rinse to remove the soap. I personally direct the water stream so that the same water passes over other dishes while I rinse the one in my hand, thus reducing the amount of rinsing needed for subsequent dishes.