Such arguments are usually emotional and spouting off facts is unlikely to change the other person's stance and instead make them double down harder.

I've attempted to engage with people in ideological echo chambers not by citing any specific facts that contradict their beliefs, but by asking them to state their own beliefs using more specific and concrete language. "What exactly is your definition of communism?", "Has Bernie made any specific statements indicating that he supports that policy? If not, what leads you to infer that he does?", etc.

The results? No engagement at all.

People do not want to be challenged in an echo chamber. Either talk to them (without convincing them of anything) outside the chamber, or walk away.

Agreed, often it's less about the person you are interacting with but about those reading your conversation.