Focus on the statement, evidence, and consequence rather than private messages, names, or identifying details. Summarize the relevant claim in your own words and replace personal examples with synthetic ones when possible.
State where you agree first if it clarifies the disagreement, then explain the exact exception or counterexample. Do not post screenshots of a private conversation to win a public argument. If verification needs confidential material, keep that part in an appropriate private channel.
The public discussion should be understandable without making anyone identifiable.