Having worked in a (very) privacy-sensitive environment, the quality of the hosted foundation models are still vastly superior to any open weight model for practical tasks. The foundation model companies (OpenAI, Anthropic, etc) are willing to sign deals with enterprises that offer reasonable protections and keep sensitive data secure, so I don't think privacy or security is a reason why enterprises would shift to open weight models.

That said, I think there is a lot of adoption of open weight for cost-sensitive features built into applications. But i'd argue this is due cost, not privacy.