Your DMs should be something you’d be fine showing the person you’re with. That’s the whole standard.

Privacy and secrecy are not the same thing, and the difference sits in how you would feel if the person closest to you read over your shoulder. Not because they are owed access, but because a message you would need to hide is usually a message you already know something about. The discomfort arrives before the trouble does. That feeling is the whole signal.

If a conversation is making you angle your screen, look at what you actually want from it. Either bring it into the open or let it go. Doing nothing and hoping is the option that ends badly.