Afaik Anthropic is not giving pretty much any provider model weights, so any inference of Opus is certainly not private. Either going through Anthropic or Bedrock, or Vertex.

Of the three Bedrock is probably the best for trust, but still not private by any means.

They do verifiable inference on TEEs for the open source models. The anthropic ones I think they basically proxy for you (also via trusted TEE) so that it cant be tied to you. VPN for LLM inference so to speak.