Duel of the Marshals.
My money is on Rooster.
Cogburn would shoot Dillon in the back with the Navy Colt, first chance he got.
Fixed typo in thread title. 
Matt Dillon had a vastly better record, so, going on form, Dillon.
In a kind of meta-argument, Gunsmoke (both radio and tv) was a bit more of a “fantasy” western than True Grit. Grit was intended to be…well…gritty. Not realistic, exactly, but at least a little more “naturalistic.” Cogburn’s eye-patch shows he’s more vulnerable than Dillon.
(Yeah, Dillon was shot in the spine once and nearly paralyzed for life…but he got over it with no remaining symptoms.)
Dillon is (to a minor degree) a kind of “superhero.” He is less bound by the laws of nature. Cogburn is not favored by the writers to the same degree.
Of course, both being lawmen, there’s no reason for them to fight…
Marshall Dillon plays by the rules and is slow and careful (in contrast to Han Solo, in the Gunsmoke opening duel, he did not shoot first).
Rooster Cogburn made up his own rules.
Cogburn once robbed a bank.
“I didn’t steal a person’s money…”
Say what?
Er, thanks. 
Rooster had only one good eye and Dillon was The Thing. Cogburn bites the dust.
If you can count The Thing, I can count the bulldozer from TheFighting Seabees.
Dillon is toast.
Even though Dillon could be shot and heal almost instantly I’ll give the advantage to Rooster for pure orneriness.
There was only one “o” in Rooster.
Radio Matt Dillon, especially in the early days of the show when it was still a rather bleak western noir, was a match for Rooster WRT meanness and ruthlessness. Further, he was among the very best of the best at gunplay. This is acknowledged by friend and foe alike. Rooster is no more than competent with a gun. Go to give it to Dillon.
Chester, and to a lesser extent Doc (who started off as a deeply cynical greedhead), were necessary to give some lighter elements to the Gunsmoke. Matt was something of a warrior monk in his sobriety and grim focus on his job. Rooster, in the movies anyway, is a drunkard and something of a buffoon. He is his own comic relief. He’d clearly rather spend his days at ease and drinking. Got to give it to Dillon again.
Of course, Hopalong Cassidy could beat 'em both at once…
He was the best. (Someone said so, in just about every single radio episode!)
John Reid (The Lone Ranger) would also beat both Dillon and Cogburn.
So would Lucky Luke…
Agreed. Plus, Dillon was just a TV marshal, not a real movie marshal like Cogburn.