The mechanism grey-market buyers have to protect their interests against over-selling bugs is tranched payments. Sellers make much of their returns from bugs on the back end through "maintenance agreements", which both require the seller to keep e.g. the offsets in their exploits current and reliable against new patch levels of the target, and also serve to cut off payment once the vendor kills the bug.
If you sell to both sides, you quickly kill the back end business from the grey market buyers. If you sell to too many or too sketchy grey market buyers, the bug leaks --- vendors see it exploited "in the wild", capture samples, kill the bug; same outcome: tranched payments stop.
This is one reason it can make sense to take a bounty payment that is substantially smaller than what a bug might be worth on the market: you get certainty of payment. Another reason is that the bounty program will only want POC code (perhaps proof of reliability in addition to just exploitability), while the market will want a complete enablement package, which is a lot of work.
...come to think of it, how does that work? Aren't the most important exploits to patch the ones being actively used in the wild?
In other words, how do they avoid someone playing both sides? "I found an exploit being used by the LEETH4X0R malware [which was in fact created by the guy I sold this exploit to] to steal people's gmail cookies."
You'd have to find out about LEETH4X0R before other researchers, but of course, you'd have a head start.
Yes they will.
If you are the murderer, there will be.
Honestly I’d be more worried about crossing the blackhats.
Security services tend to anonymously report security flaws they use after use against any high value target, since they don't want the opponent using those same flaws back at them.