In Squid Game, we see that a standard method of recruitment for the titular lethal competition is playing an extended game with candidates in highly surveilled subway stations, which seems to involve a priori suspicious activities such as publicly and repeatedly slapping a well-dressed businessperson in the face, after which they actually hand out their business cards. Presumably, many more people go through these stages than actually join the game in the end.
We see that the organization is perfectly willing to let players go if a majority vote no, and we see this happen. We also see that they make little effort to prevent players from telling the authorities, which in fact does directly lead to a police officer investigating. Especially since they drop players off together, they certainly could not rely on everyone making wild and uncorroborated allegations as Seong Gi-hun did (as opposed to two people saying that a well-dressed man promised them money then kidnapped them, say). Certainly, they have little means of preventing people from telling their friends or family.
We also see that by the end of each year's competition, they kill a number of people that seems to be equal to a non-trivial chunk of South Korea's actual adult missing persons cases.
Further, the guards are sufficiently poorly vetted that they can tell international organ traffickers about the location of the island and hand off organs to them without the organization noticing. Finally, we see that they let one person go every year with sufficient money and reasons to potentially cause some trouble for them.
With all these potential leaks, how do the organization and its games stay secret?