Start by listening
Most arguments about the news are really arguments about values, experiences and fears. Asking someone what worries them about an issue, and listening for the answer, tells you far more than a counter-argument would. People relax when they feel heard, and only then can they hear anything back.
Agree on the facts first
Before debating what should be done, check what you both think happened. Often the disagreement is smaller than it seems, or it comes from two people reading different sources. Saying 'where did you see that?' is more useful than 'that is wrong'.
Ask questions instead of making points
Curious questions such as 'what would change your mind?' or 'what do you think the other side gets right?' open a conversation. Rhetorical questions and sarcasm shut it. The goal is to understand, not to win.
Share sources, not just conclusions
If you want to challenge a claim, offer the source and invite the other person to share theirs. Reading each other's sources, even briefly, turns a clash into a comparison.
Admit what you do not know
Saying 'I am not sure' or 'I might be wrong about that' is not weakness. It makes the other person more willing to do the same, and it is simply honest, since most news stories are complicated.
Know when to stop
Some conversations are not going anywhere, especially online. It is fine to say 'we see this differently' and move on. A friendship or a family dinner is usually worth more than winning a point about a headline.