To answer the question as asked, I have the thoughts of the POV character in italics, no matter who's speaking or whatever else is going on. One time I had a character tailing a pair of non-perspective characters in conversation. The character can't see them as they're talking and has to focus on moving around silently, so it's very much limited to his perspective, as the only visual window into the story is his eyes. It was not a drawn-out conversation by any means, but avoiding an overlong conversation, whether eavesdropped or not, is a good rule in general. I suppose I should put this in an author's note somewhere, but thoughts in my stories are assumed to take place instantly, so the insertion of the perspective character's thoughts, to the tune of- That explains why you were lying earlier- in response to a revelation from a talking character does not lengthen the time that passes in the scene. Having the thoughts of the POV inserted into the dialogue as it's taking place seems to remind me at least that we have access to the thoughts of that character and only that character, which is not the case with omniscient narrators.
I'm confused as to why you find your conclusion unsatisfying. Did you wish to write a really long conversation to be overheard by the perspective?

 

You could make the overheard conversation only partly understandable. That way the POV character gets periodic need to interpolate what was heard with some evaluation of meaning; it's like a short mystery puzzle. The scene involves incomplete information, so both POV character and reader must complete it. The trick is finding key pieces of the puzzle to remove that wouldn't invalidate the overheard conversation.

A version I used, however brief:

 

Stuff I'd put in, interspersed with overhead dialogue:

1. moving closer to repositioning to hear more-- "Harry snuck closer, taking care that his sneakers didn't scuff and give away his position"

2. internal dialogue on what the sneaking character thinks of what he's overhearing-- "So that's why they've been avoiding me, he thought furiously. Ron and Hermione are off buggering away while I'm stuck with nothing but Fred and George's toy wands! "

3. hard to do well, but sometimes fragments of dialogue can work:

"Hagrid said ---dog-- don't know where" Ron said, and Harry struggled to hear the rest. A dog? Why would Hagrid care about a non-magical, non-scary dog? He already has Fang.

Fragments of dialogue can get annoying to type out and read sometimes, but I've seen it done well occasionally.

 

It depends on whether your charecter can see or just hear the conversation. Personally I think things to keep in mind are:

What information absolutely needs to be conveyed and how can it be conveyed in as little direct speech as possible?

How can you make it not seem like it is super convenient that the eavesdropper just so happened to hear exactly the information needed?

How well can the eavesdropper hear the conversation, can you skip non-relevent sections by having the sound of their voices fade/be covered? Are they trapped for a long period of time and tune out of the boaring stuff untill some phrase or name jerks them back into listening?

Once you have trimmed it to the nessasary parts, you can fill it out with a number of things

• What noises etc, give the listener clues as to how the conversation is playing out/being reacted too (pacing/crying/tone of voice etc.)
• How the listener reacts to the news.
• Whether there is any tension about them getting caught eavesdropping.
• Whether they can do anything (perhaps taking risks) to hear better or see the conversationalists.
• Whether anything can get misconstrued.

At the end of the day, it's a rather blunt force plot device and should be used sparingly, but it's common for a reason and has its own utility.