Chinese whispers is another name for the game also known as telephone, grapevine, or whisper down the lane. In this game, each person whispers a phrase to the next person in line. The phrase is usually unrecognizable by the time it reaches the end of the line.
"A chain of Chinese whispers" appears to be used for the analogous real-life process where a rumour gets totally distorted while being passed along.
Per my first comment to the question, "guesstimates" from Google Books: one person's word against another's -4990, one person's word against another - 3120. So you could say the "full" version "wins". But they're only estimates which are often wildly inaccurate - on a more specific search where I can actually check them all, it's another - 27, another's - 21.
The logical position - since it's always "your word against his" (not "your word against him"), obviously OP's version should also use the possessive.
But in practice many (perhaps even most) writers apparently choose to ignore that. And it's a pretty safe bet they're even more likely to do this in speech (people tend to write more "correctly" than they actually speak). And I rate the "grammar" of speech higher than that of logic and textbooks.
I personally find the "logically/grammatically" correct version unnecessarily cumbersome. There's no case for claiming the meaning could possibly be affected - we all know what it means, and it would be perverse to suppose "you against him" means anything different in such contexts than "your word against his word".
TL;DR: Pedantically speaking, you "should" use the possessive apostrophe. But I wouldn't bother.
As it happens (and as I just commented to @David's answer) I noticed "I asked her for a list of Charles' friends" when watching Missing, 1982 last night. Only one "s" was enunciated, which seems perfectly normal to me. If the context already strongly implies the possessive, why throw in another awkward consonant? So many people don't, it just seems pointless to claim they should.
Best Answer
It's certainly not what I would call an idiom - a group of words established by usage as having a meaning not deducible from those of the individual words. Google Books has just 8 instances of “another link in the daisy chain”, and most of those are this Lolita quote. That's as opposed to 143,000 for “another link in the chain”.
The word "daisy" in OP's version probably doesn't add any meaning - it's only there because the writer is aware of the actual idiom daisy chain, which as a noun can mean a series of associated people or things, and as a verb can mean to connect (several devices) together in a linear series.
In general, chains are strong (though in the context of a chain, a link is usually the weakest link). Daisy chains, on the other hand, have no connotations of strength - they simply convey the interconnectedness of the "linked elements").
Note that later in the text, Nabokov writes "I still have, vibrating all along my optic nerve, visions of Lo on horseback, a link in the chain of a guided trip along a bridle trail". I personally do not think it would make any difference to the book if he'd swapped "daisy" to that sentence, used it in both, or left it out altogether.