Private cloud computing is not just trusting who runs the code (cloud provider) but also who wrote the code (service provider).

Who wrote the code is the bigger source of threats, because by design it could use your data in ways you don't want.

Whereas the one running the code obviously isn't supposed to muck with the code or data, and would only do so if compelled to do so or by accident.

That's why private cloud enclaves put more focus on the service provider than the cloud host.