My opinion:
Firstly, in WB vs WG, Mihawk slashed at WB, Jozu can hold against it. Jimbei punched Moria hard. Jozu punched Crocodile (due to Jozu's immense strength and speed, or haki, Crocodile hasn't reacted yet to change into intangible sand). Jozu was controlled by Doflamingo, but not fully. These events implied that though there was difference among Shichibukai's strength (and also Jozu), but the differences is detailed and depended on fruit match-up and natural opponents. They can trade blows with each other, which is why i think Shichibukai members are standing on almost equal level. And which is why i think Strawhat crews aside from Luffy, will reach their full potential at Shichibukai level. Which is why i think Zoro will be Mihawk's equal and beat him eventually, when he reach that Shichibukai level. I know the level itself varies, but the range is not that wide. Blackbeard is an exception, due to his likely possibility the Last Boss candidate, so like Luffy, he is STILL growing.
The Yonkou is a different matter. The last dungeon in the game, the last stand is The New World. Coby said "The one who conquered The New World will be the Pirate King". Moria said "...Then see for yourself what a true nightmare is...in The New World!!". No matter the crews, the leader is strongest by a significant difference even to the first mate. Marco is strong as hell, able to match a Logia Admiral. But Whitebeard is completely beyond it. There is no reason for Whitebeard NOT to beat up Shanks and make Shanks' crew as his own, if Shanks is as WEAKER to him as much as a LOGIA is WEAKER to him. There is no reason NOT to take one of the Yonkou to become his first mate, for example, rule half of and eventually conquer The New World. The ONLY reason, even if we substract all of both sides' crews, is that Whitebeard and Shanks are equals. That is the level Luffy must reach. That is the level ONLY Luffy, in Strawhat Pirate, will eventually reach. That is the level of Yonkou.
Mihawk often sought Shanks for duels. This shows that Mihawk’s chased Shanks. I'ts been showed that Shanks was the one being annoyed because he thought that Mihawk would challenge him again, "I'm not in a good mood now. Are you here to fight me?"
Mihawk’s cool challenge-loving character – displayed the most at the end of his “fight” with Zoro – shows that he wants people with enough resolution and will to rise and become his equal. He’s willing to wait for Zoro. "Be stronger, kid, much stronger, then try to beat me". This is "waiting" attitude directed towards Zoro, while Mihawk is the one being challenged.
The way I see it, the two "attitude"s are similar. Zoro towards Mihawk and Mihawk towards Shanks. The difference is in the personality of the ones being challenged, Mihawk ("okay then, but try to be stronger first, kiddie") and Shanks ("dammit, you're gonna ask me for a duel again?").
I'm not equalizing. Strength difference of Zoro to Mihawk is clearly wider than Mihawk to Shanks, but from previous explained duels and attitude towards the other, it's implying to Shanks being stronger, because he's being challenged, not the way around.
Mihawk previously sought duels with Shanks and all. Another example, In WB vs WG, he wanted to know the difference in strength between himself and WB. So he launched an attack. A chasing attitude, towards a stronger beast. This treatment, to me, was somewhat similar to Mihawk’s chase towards Shanks. That Mihawk saw Shanks as someone he must surpass. As in someone more powerful.
But no, not someone more powerful in swordsmanship. It is clearly stated that Mihawk is the strongest swordsman in the world. In the sword discipline, he’s numero uno. Shanks carries around a sword, he can attack using a sword. It does make him a swordsman, I think. But is he someone who follows the way of the sword by his body, mind and soul, like Zoro or Mihawk? I don’t think so.
Zoro uses ALL of his other skill to support or strengthen his swordsmanship. Mihawk does, too, from the way things being stated. That's, i think, the true meaning of "swordsman" in that title of Mihawk's. But Shanks, very likely, will be different than those two.
In my opinion, Shanks will be shown to us as someone more like Brooke, a swordsman but whose strongest ability is something else which stands by itself. Whitebeard, who holds a bisento, obviously can attack using a bisento. Is he the strongest bisento fighter in the world? Maybe, MAYBE NOT. But he has his hack Devil Fruit ability (and really possibly Haki also). Whitebeard's ability can stand as its own, NOT to support his bisento-playing skills. So is he, counting all aspects, the strongest man in the world? Absolutely. THIS is the difference.
Shanks may have swordsmanship that is second ONLY to that of Mihawk’s. He may have swordsmanship that is ALSO beatable by Zoro’s “Listen to the breath of things, cut nothing, and cut anything” swordsmanship. He may, though it’s less likely, have swordsmanship that is ALSO beatable by that of Tashigi’s. We don’t know for sure.
What we do know is, everything said and praised about Mihawk had been solely about sword. “Oh, he diverted those bullets like nothin’ by using his sword.” “Oh, he split open the galleon like nothin’ using his sword.” “Oh, he’s the goddamn strongest swordsman in the world.”
And something else that we do know is, Marco and Jozu and Whitebeard didn’t mention or gave compliment to Shanks’ swordsmanship at all. They didn’t warn the others, “Oh, it’s the second-strongest swordsman in the world. Be careful,don’t come near him or his sword will stab through your butt before you got the chance to crap yer pants.”.
Instead, they emphasized his Haki greatly. It may or may not be his strongest point, his Haki, but it is clearly more dangerous to Whitebeard’s crew than his sword. Therefore I can conclude that Shanks’ swordsmanship is not his main weapon.
Shanks’ ultimate power, his Haki or something else godlike that’s for sure, or their combination, is what made Mihawk sought after him for duels.
From Mihawk’s battle-hailing personality, he won’t judge someone as weaker than him like, “Ah, dang, I’ve chased him all this time and he lost an arm, now he must be weaker than me. It’s no use chasing him anymore” No. True warriors do not underestimate their duel target. True warriors, no matter the result and no matter how hard they themselves want to win, want both sides to fight as the fullest. Knowing Mihawk, he’ll think like this, “Ah, dang, I’ve chased him all this time and he lost an arm, now I’ll never know, never feel the extent of his true power. It’s no use chasing him anymore.”
My conclusion: Mihawk beats Shanks in a sword fight. Shanks beats Mihawk in a fight to the death.
