I constructed "we both know men" which does have a pretty different meaning from "we know both men"
Could this possibly be translated as "we both know men" or would it be more like "we both know about men"
Znamy obu męzczyzn. . We know both men. We know Adam and Marek
Obie znamy mężczyzn - We both (Ania i Kasia) know men
Obaj znamy mężczyzn- We both (Adam i Marek) know men
Oboje znamy mężczyzn - We both (Kasia i Marek) know men
I think those three at first interpretation are about "we know how men are like".
Frankly I would translate this to English as "We know both the men" or perhaps "We know both of the men", but I am not a native speaker of English, so I do not know whether it would be the best translation possible.
I wrote "We know both men." You could add "of the" but just added "the" sounds like some of the dialects that often just ignore "of" before "the." I think only a few dialects do that but I am in one of them so I don't really know how common that is.
And why is it "mężczyzn" and not "mężczyzni"? I thought it was accusative (biernik)
The word "obu" is accusative of the male-personal (m1) numeral "obaj" (meaning two of two available). It is a bit special one, because there is none similar for other quantities ("three of three available" etc.), but it works similar to collective numerals. It means that the noun following the numeral takes the same case as numeral, in plural. And "mężczyzn" is right accusative plural of "mężczyzna"
For a simple explanation about usage of polish numerals you may take a look at wikipedia article you may also like to take a look into Polish Grammar in a Nutshell starting from page
553 39, or to an in-depth study here