As a DM do you punish players who are using stealth, but still talking to each other at the table?

No, part of the fun is people talking to each other. Although I will “punish” if it’s funny.

It’s not “punishment” to run the game consistently.

I don’t watch “Dragon’s Den”, so I can only imagine what you’re talking about.

But if people say they are trying to be stealthy, but then start jabbering at each other, then yeah there will be noise, and if there’s someone near enough to hear and notice it, they might.

That’s a pet peeve of mine in many shows where people are supposedly trying to sneak around, or even out-maneuver armed foes in the dark, but they’re stopping to loudly chat, look at each other, shine flashlights around, etc.

I’ve run dragons den encounters before. Sure, they use stealth and that’s fine. I have them roll a stealth and I have the dragon roll a perception. That’s how I run it. I don’t mind if the players talk to each other. Although I will object if I have one player try to tell another player how to run their character, especially if they’re in different parts of the room or whatever.