The don't say it's instantaneous but they do say it's automatic. There's a huge difference between "You know [contacts etc.] ..." and "You might know [contacts etc.]...".
Knowing is automatic, by RAW.
Finding the messenger and getting them to agree to carry your message is not. For that matter, considering that two of the three types of messengers suggested are people who have jobs that keep them on the move, the local messenger may not even be there.
But that's only if you care about RAW. As some people have claimed, this is such a niche feature, it may only come up once or twice in an entire campaign--so why not throw RAW to the wind and let the criminal shine for that moment?
(I swear, people are treating these features like they're going to be a constant thing that can be used to screw over every one of the DM's plans.)
By RAW the DM is free to say the first and allow the player to self-limit the character, but not the second because that imposes a roll; and as written, success in finding a contact is automatic.
But see, here's where you can go against RAW and still allow the player to use their ability. Require rolls. Require roleplaying. Require the PC to put in the effort to find and befriend a messenger. Allow the PC to get in touch with the contact through other means.
However, the
actual point of the feature, as I said to mamba, is that the contact is a "liaison between you and a network of criminals." In other words, "you know a guy who knows a guy." The messenger is just a way for you to get the contact's attention.
For any other character, if they want to get in touch with, say, an illegal poisons-dealer, they would have to do the legwork themselves and potentially run afoul of the law or other shady characters who are interested in knowing why a group of travelers suddenly want an illegal poison. But the criminal can just say "lemme make a call" and your contact then gives you an address.