Introduction
There seems to be a pervasive misunderstanding that when Shanks had both his arms, he was equal to Mihawk. That they duelled several times, and the results of their many duels was a draw. As far as I can tell, this misunderstanding is based on mistranslations and personal headcanon; the manga does not at all support it.
In this thread, I'll try to set the record straight. I'll be using the Viz translations throughout the post.
A Single Duel
In the first place, there's no actual indication that Mihawk and Shanks had multiple duels. There's no reference to it anywhere in the manga.
Indeed, when Whitebeard met Shanks, he referred to just
one duel between Shanks and Mihawk:
Insomuch as references are made to Shanks' fight with Mihawk, it's to this lone duel.
Edit
@Nidai_Kitetsu pointed out that Stephen Paul (Viz's current translator) translated it as "multiple duels" not a single one:
(I guess this fits better with Shanks asking Mihawk if he intended to challenge him.)
An Unresolved Battle
Another misunderstanding appears to be that Mihawk's duel with Shanks was unresolved or at least ended in a draw. As far as I can tell, this is due to a mistranslation. When Mihawk visited Shanks, there was no reference to "settling" things/the score.
Instead, Shanks simply asked Mihawk if he was here to challenge him and Mihawk replied that he did not challenge "one armed has-beens":
So there's no actual suggestion that the duel was undecided/unresolved. Indeed, it's likely that there was a victor.
Who was the Victor?
If I claim that there was a victor of Mihawk's duel with Shanks, it's only natural that you ask me to defend this statement. Fortunately, the manga is pretty clear on what the outcome of said duel was.
Brannew said that Mihawk's swordsmanship exceeds "Red Haired the Emperor" (Shanks):
How would Brannew know that Mihawk is the superior swordsman if he didn't win the duel? If he drew their duel, there would be no basis for elevating him above Shanks. Indeed, the very fact that it is Mihawk — and not Shanks — that is considered the World's Strongest Swordsman already heavily implies that Mihawk won their duel:
Further corroborating this manga statement is Mihawk's Vivre Card that states that he's waiting for a swordsman who surpasses Shanks:
This reinforces that Mihawk has surpassed Shanks and no longer considers him a worthy challenger.
Conclusion
The claim that Shanks drew his duel with Mihawk — or that it was otherwise unresolved — is simply without any basis in the manga and is indeed nothing but sheer cope and wishful thinking.
