I guess it might depend on where you live: mine’s a forested area, so not much dust. Never been one to worry a lot about small amounts of dirt either: when I pull a fresh carrot from the ground, I never bother to wash it, just wipe it with two strokes on my sleeve or trousers before I chew down!
So, dandelion and other wild field greens require extensive cleaning, if you have a normal desire for cleanliness and not a geophagia pica.
But you’ll miss a lot of minerals! 
(Admittedly, most of them are insoluble, so they won’t serve you much. But there are also some beneficial bacteria!)
If you pick the dandelion greens and hold them in a bunch in your hand, then tie the bunch together with a rubber band before placing in your basket or bag, you’ll find them much easier to clean back at home - you can hold the whole bunch and swish it around/up/down in a deep bowl of clean water - the dirt that comes off will sink without falling onto other leaves.
Once the bunch is washed, cut off and discard the bottom inch where the bunch is too tight for the water to get in, then give the loose leaves a final rinse in new water.