Its part of my buddy checks, gas quantity and set on DC. I say aloud, check mine and buddy acknowledges and check theirs.
I dont see a lot of thorough buddy checks out in the wild though.
Others will have said it.
As a buddy pair, you should know each others limits for the dive. Crucial to this is an understanding of how deep you are each happy to be, for how long, and how much decompression you will run (or the 'No Stop' dive limit).
Inherent in this is you need to know what each of you is breathing. As the Nitrox diver, you need to know that the dive time and decompression is dictated by the air diver (or diver on the weakest mix). As the air diver (or weakest mix Nitrox diver), you need to acknowledge that the maximum depth of the dive is dictated by the Nitrox diver. This a fundamental part of all Nitrox courses.
It is the nature of my diving that I now often dive mixed teams with mixed breathing mixes. This is because I am on CCR, and most of my buddies are on OC.
Time and depth is normally dictated by my buddy. As such, one of my computers is normally set to the mix my buddy is breathing. Especially if they are an unknown, or a 'newer' diver.
The issue that limits my depth, is what's in my DIL, and what Bailout I am carrying. As a general rule. If I''m using Air in the Dil, MOD is around 40m. My 'recreational' Bailout is normally 27%. Which is a MOD of 42m (at 1.4 PO2).
As a dive pair you made a mistake. Nothing bad happened. Learn from it, and avoid the same error in the future. You should have been suspicious when his computer cleared when yours did, he should have ensured that he had the correct mix set.