Are the contents of the overheard conversation at the beginning of the game ever resolved?

Solution 1:

It turns out it wasn't supposed to mean anything, or if anything, it was there to add a bit of paranoia to the relationship between you and Delilah (a recurring theme in the game).

IGN did a spoiler podcast with the developers of the game and brought it up (at 34:50):

Marty Sliva (IGN): ...like the way you react to Delilah: whether you call her out on her weirdo conversation with someone else. Which also, I want to talk about: what was the deal with that?

Sean Vanaman (Campo Santo): Yeah, people freak. She was just talking to someone else.

Marty: Why was she talking about me?

Sean: She wasn't talking about you.

Jake Rodkin (Campo Santo): You just assume that if anyone's talking it's about you?

Sean: You go back and listen to it, she just wasn't talking about you.

Mitch Dyer (IGN): Yeah, the second time through it's very clear. She's having a completely separate conversation. And if you ask her about it and be like "hey, what's up?" "Dude, I'm in the middle of something: please leave me alone. I'm just talking to someone. What are you paranoid about?"

Sean: It's funny, if I had to do like a director's cut version of the game right now, I would probably put in a third choice there, where you can say, "hey, uhm, I kinda heard you talking to someone. I'm sure it was nothing." You know, just like, the more pokey one? And she'd be like "oh yeah that was just this dude in another lookout tower and we're just dealing with this bureaucratic mess and I don't know what this guy is doing but it's annoying. Anyway, you ever have any coworkers?" And you'd just be like "oh okay, yeah, that's life."