How do I obtain the secret "Chatterbox" achievement?

I, too, hadn't heard of this achievement until just now. Luckily, PowerPyx over at Xbox360Achievements had. He even put together a nice list!

During the course of the story there are 7 occasions where you can talk to your friends. You will have to listen to everything they have to say. If you miss one single conversation you cannot get this trophy / achievement in your current playthrough anymore. You need to keep talking to everyone until the Square X symbol above their head is gone. You will also see the TR auto-save logo flashing in the bottom right corner of the screen when all conversations with one person are done.

You need to talk to the following people:

• Before searching for 50 pieces of salvage, talk to Whitman.
• After finding the 50 pieces of salvage, talk to Whitman again (do not repair the Axe yet!)
• After helping Roth (where the wolves attack him), talk to Roth.
• Shortly after getting the Rope Arrows you will meet Roth again. Talk to him before using the zipline.
• When reaching the shipwreck beach you will meet your friends. Talk to Sam, Reyes and Jonah.
• Before going to the Endurance to help Alex and get the tools, talk to Whitman, Sam, Jonah and Reyes at the beach camp. Reyes is on the boat.
• After coming back from the Endurance, talk to Whitman, Reyes and Jonah.

He's also detailed the entire thing on YouTube.

One thing worth mentioning is that, if you have 50 salvage before speaking to Whitman the first time, it should give you all of the conversation options before you upgrade your axe, meaning you would be finished after clearing all of the dialogue with him (the chat option stops appearing).

Unfortunately, the achievement is missable, meaning you won't be able to get it without playing through the game again (assuming you've already missed one or two, that is).