With OpenRouter this is pretty easy to see - each model page has various providers listed along with an icon marking providers with zero data retention, and the price. You can also set up an account-wide "guardrail" to prevent routing to non-ZDR providers.

OpenAI and Anthropic also claim they don't retain data. I'm not sure what it would take for me to trust any of these inference companies, and why I should trust them more than the big ones.