Both Watches, and the Pikes, guard their territories jealously. More than one criminal has escaped punishment by crossing jurisdictions. Too, more than one has drowned trying to swim into a new jurisdiction.
The White Watch is both better equipped and less prestigious than the Black Watch. The Merchants with an interest in the Grossemarkt make sure that those keeping their warehouses safe are adequately prepared for the job. The side effect of all this is that the White Watch is much more concerned with crimes of property then anything else. "Life is cheap in the Grossemarkt, everything else is dear."
The Black Watch is more numerous and more ostentatiously present. They are interested in keeping the streets safer, since the inhabitants of their beats tend to be middle class and up. "Shallya have mercy on the footpad caught in the Uberstadt, for the Black Watch certainly won't."
In the Rechtenufer, they are a bit more lenient; boys will be boys, after all, even if the boys are university students brawling with gunnery students. As long as they keep things amongst themselves, the Watch turns a blind eye. Woe betide them, though, if the youthful hijinks don't stop at the Altebrucke or inconvenience their elders and betters.