Exact numbers are pretty loose, but here goes... Each uncoated air-glass surface scatters 4 to 8% of the light passing through. The front element reflects it back out into the world, and the rear surface only reflects what's bouncing back from the film plane, so I discount those two surfaces. So an old uncoated Dagor with two inner air-glass surfaces is losing 8-16% of the light, about half to flare and the other half absorbed by the barrel walls. So you're talking 4 to 8% flare vs. clean image making light. (BTW, two inner air-glass surfaces is the minimum with compound lens designs, so the Dagor was very popular because of its high contrast.)
Split the Dagor into an "air-spaced Dagor" (aka Plasmat), and there are now 6 inner surfaces, so 24 to 48% of light is lost, with 12 to 24% being flare. That's why the Plasmat was never popular until coatings came in.
Early coatings reduced flare to 1 to 2% per air-glass surface. Modern multi-coatings are claimed to get it down to 0.1%.
The flare acts like pre-flashing the film. It affects the shadows most because the flare is evenly distributed, while little image-making light is hitting the shadow areas, so the flare can overwhelm it. If you adjust your processing for uncoated lenses, coated lenses will seem harsh, and if you adjust your processing for modern lenses, uncoated lenses will seem flat or mushy.
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