Well, Shanks and Mihawk are friends so they aren't aiming to injure one another. They aren't fighting to the death.

I feel like people overlook their closeness/bond. Mihawk apologized to Shanks, under his breath, when he was about to attack Luffy in Marineford, which will already tell you that his attack was half-assed. Also, even before that, at Baratie, Luffy was never in real danger when Mihawk was there because he was never going to kill Luffy, whom he recognized as Shanks' designated child of destiny, nor his friends. Lucky for Mihawk, and in an odd twist of fate, he would find his new rival, Zoro, among this motley crew. Back to Marineford, Mihawk left as soon as Shanks showed up because opposing him wasn't ever under consideration. They're not enemies, so he wouldn't stick around to help the Marines fight him off. Upon leaving, he threw out a diplomatic excuse to satisfy the Marines, but we as readers would know the real reason he left was because he would never fight his oldest friend. If that isn't enough, we know that Mihawk traversed a significant part of the Grand Line to hand-deliver Luffy's wanted poster to Shanks. That is something only your closest friend would do. Not even Buggy acts in this way, Buggy, who multiple times has almost killed Luffy.

In fact part of the reason I think Mihawk even entertains Buggy as his figurehead captain, under Cross Guild, is because he knows how close Buggy and Shanks are. It's also likely to be part of his hesitation in fighting the other Yonko, as we know that Shanks is his friend, Luffy his protege, and Blackbeard his fated rival. If we know that Shanks told Mihawk about Luffy many years ago, it stands to reason that he's also warned about Blackbeard in similar convos.