The Convenient Eavesdrop is my name for a trope that's particularly common in fiction for younger readers. (There's probably a TV Tropes name for it, but I dare not risk my productivity by going to that site.) It's when someone just happens to be in a position to overhear someone else, usually the antagonist, discussing exactly the information they need to know to move the plot forward.
Enid Blyton's Famous Five had frequent Convenient Eavesdrops, to the point that the parody Five Go Mad in Dorset had the villain's minions muttering, "Rhubarb, rhubarb, secret plans, rhubarb, rhubarb." In their case, generally they'd overhear something that revealed Dastardly Doings Afoot, and either tried to tell the adults and weren't believed, or didn't tell the adults because they wouldn't be believed. The Convenient Eavesdrop functioned as an inciting incident, something that would get the adventure underway, because it was now up to the kids to resolve the situation.
Convenient Eavesdrops can also kick off the plot by being misinterpreted. This is often, though not inevitably, comic; Richmal Crompton's William books used it for great comic effect, as have many farces and sitcoms, but it can also lead to tragedy or near-tragedy when someone acts on misinterpreted information that suggests their lover is unfaithful, for example. The misinterpreted Convenient Eavesdrop is a sub-trope, rather than a trope flip or subversion, since it's been around for so long.
More of a flip, though it's also been around for a while, is having the antagonist conveniently eavesdrop on the protagonist. Because you'll probably (though not certainly) be in the protagonist's viewpoint, you then need the protagonist, or a companion, to notice the antagonist sneaking away and realize that they've overheard. But what have they overheard? How much do they know? Can we contain the situation? Will they blackmail us?
Another use of Convenient Eavesdrops is common in the Harry Potter books. Apart from the prologue to each book (it's called Chapter 1, but trust me, it's a prologue), the books stick tightly to Harry's point of view, so he has to find out any other relevant information that explains what's going on by either being told it, or overhearing it in a Convenient Eavesdrop. There's a particularly blatant example in Deathly Hallows, where Harry has been randomly teleporting around Britain, on the run, and doesn't know what's happening. Super-conveniently, one of his school friends, a goblin, and a third person I forget just now randomly happen to pass nearby while Harry's hidden in a bush, discussing exactly what he needs to know to get the stalled plot moving again.
You'll have gathered that I'm not a fan of the Convenient Eavesdrop, which I consider a close cousin to Deus Ex Machina (the convenient event not triggered by the protagonist which saves his or her bacon at the critical moment, because the author has written themselves into a corner). But the problems it solves - knowing what the antagonist's plans are, or otherwise getting key information the viewpoint characters have no other access to - are genuine problems. Are there better ways to solve them?
I think so. I think the solution is the more general solution to other problems of protagonist agency versus coincidence: instead of having something happen by itself, have the characters cause it deliberately.
Let me offer an example from Auckland Allies, the first novel in my urban fantasy series. (I'm writing this blog post as a way of warming up to tackle revisions on the second book.) Mysterious occurrences are occurring; hostile black-clad men keep turning up and trying to harm the protagonists. The protagonists want to know why.
The lazy way would be to have them in the right place at the right time to find out by Convenient Eavesdrop. But the other way to do it - which ends up driving the plot for several chapters, because it gives plenty for the characters to strive for - is for them to set out to find out deliberately.
During one of the pursuits, Sparx, the technomage, used a cellphone to shoot video of the SUV pursuing him and Steampunk Sally. He's able to read the license plate.
They're not official law enforcement, so they have to find a way to trace the plate to an address. He and Sally run a scam on a used-car dealer to get access to the website where they can do this.
They go to the address and observe. Sparx finds an active wifi node, and attempts to hack it. Not much success, but he gets some information they can use.
Using the partial penetration he achieved of the antagonists' network, Sparx then manages to get into their webcam using a combination of magic and technology. This reveals a clue pointing to the antagonists' plans, which the Allies now set out to foil.
All of this is, I think, much more satisfying - and certainly gives a lot more plot - than happening to be in the right place at the right time. This example uses technology, which opens up more options (particularly since I have a character who's very good with it), but you can still take this approach in a non-technological setting, or with less skilled characters. Build tension as the protagonist sneaks into the antagonist's lair, past dangerous defences, and conceals herself where she can overhear the secret meeting - somewhere cramped, perhaps smelly, where she'll be vulnerable if discovered.
This is the Difficult Eavesdrop.