If its 2008 again and I had to select one player it would be Josi. He never had the same supporting cast that Dougthy had on those LA cup winning teams. If he had, he would likely have a ring too. Josi was basically the sole star player Nashville had after Weber left while Doughty was part of an ensemble in LA. And he, Josi, had much better longevity than Dougthy as well. Doughty is better defensively though and a better clutch player as well.
Josi has certainly aged better but that doesnt mean he's had better longevity than Doughty.
Doughty came into the league as a 19 year old, and was a Norris finalist the next year, then got ample Norris votes every year until the age of 28.
Josi didn't start getting serious Norris consideration until 24, and it's continued until his current age, 33.
So both have a 9 year span of excellence. Josi may add to that but he's also at the age were a sharp decline is becoming more likely. So as of now they've been elite for about roughly the same amount of years.
Here are their Norris finishes
Roman Josi: 1, 2, 2, 5, 5, 7, 7, 11, 14
Drew Doughty: 1, 2, 2, 3, 6, 7, 9, 10, 11, 18, 19
It's pretty damn close.
I'm too lazy to do it but a closer inspection of their competition during their best years might sway the case for one or the other.