Does Uchimizu (watering streets in summer) help reduce temperature in an area? It appears that in Japan, there is an old custom called Uchimizu. The name is a combination of words "road" and "water". The essence of it is moisturizing small parts of populated areas like sidewalks, streets, parks, gardens. A very nice diagram explaining how Uchimizu is performed can be found on this site.
Apart from being a traditional and cultural thing, various sources (Wikipedia) claim that it helps reduce the heat in the surrounding area, but don't say by how much exactly.
They also say that it prevents dust particles from spreading, but this is a more obvious part of Uchimizu, as watering sandy ground should prevent the wet sand particles from flying around. I'm more interested in the cooling effect of Uchimizu here.
Let's say we have enough people watering everything in a small city, enough to make 1/2 of all the outside surfaces wet at all times. Potential harm to structures aside, by how much degrees Celcius will the average temperature in that city decrease? Let 30°C be the average summer temperature during the day.
What about watering only the area in immediate proximity? For example, everything around your house. Will that help reduce the summer heat?
 A: It's done outside many smaller stores near where I live (Tokyo) and has zero apparent effect on the temperature. Midday summer weather is high-30s temperature and rainforest-level humidity, so the half-bucket of water tossed out the front door does little except moisten the pavement. 
A side effect is it tends to make the place messier - dust plus lots of water creates clean sidewalks, dust plus not enough water creates muck, which is harder to clean up. 
Thunderstorms (or typhoons) will cool the place down but the day after produces even higher humidity as all that water now evaporates under the clear skies that follow the weather.
The "tradition and culture thing" you reference is the real reason. Uchimizu might have made sense in the days of dirt streets but today it's just about appearances.
A: Sprinkling water on the sidewalks will cool them down in exactly the same way that sweating cools you. In both cases it is due to evaporative cooling. The same idea has been used for millenia to cool water in hot climates.
The only downside is that it will increase the humidity of the air, and humid air feels hotter than dry air because humidity slows evaporation of sweat.
As for how big the cooling effect is, I think it would be hard to calculate that. The latent heat of evaporation of water is 2.3MJ per kg, so every kilogram of water that evaporates will remove 2.3MJ of heat from the sidewalk. How much this will decrease the temperature depends on the specific heat of the sidewalk and the rate at which heat flows back into the cooled sidewalk. Presumably the effect is significant otherwise people wouldn't do it.
The energy in bright sunlight is about 1kW/m$^2$, so to balance out the energy from the sunlight you'd need to evaporate about half a gram of water per square metre per second.
