So far I've seen this most commonly in the UK. In fact what I often see is that people in the UK use a tub of dirty water with soap in it in their sinks, just dunk the dish in it a few times and go over it with a sponge, then put it to the side to dry. Rather than having clean, running water to go over it continually while scrubbing with a soapy sponge then rinsing the soap residue off with said water...
Disgusting tub of dirt dishwater as a way to "clean" dishes aside, I can see some logic to people who rinse the dishes before use - as I know my mum regularly does this (even though she washes dishes like I do with rinsing them thoroughly after washing) because a lot of her dishes are not kept in a cupboard but out on open shelves in the kitchen and so tend to get dust and dog hair (!) on them just by virtue of being in the house and not in an enclosed space.