There's a big difference between general "emissions" which a lot of people take colloquially to just mean greenhouse gases (and CO2 in particular with combustion engines), and then the small particulates and other things (like NOx) that create worse local air conditions[1]. Most "modern" diesels in use do still have the latter problem. I think at the very cutting edge, if you maintain a very good filter and operate in favorable conditions you can cut those emissions dramatically, but as far as I know it isn't widespread among the personal diesel fleet.
Most local air pollution, counterintuitively, doesn't even come from exhaust.
Wear on tires, brakes, and the road each contribute more individually.
There's another process called resuspension that dwarfs even these. Cars grind down large particles sitting on the road, making heavy particles light enough to float in the air. Then the cars' wake kicks those newly light particles into the air, suspending them in the air.
That process is driven by vehicle weight, size, and aerodynamics, and contributes about two-thirds of the particulates from all vehicle sources.
One of the interesting takeaways from all that is that average vehicle weight can be massively important to local air conditions (no pun intended).
I think there's still some debate about the exact relationship between weight to PM output, but lighter frames and smaller vehicles are probably a good idea:
[1] http://theconversation.com/fact-check-are-diesel-cars-really...