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not really

If your connection to your private cloud passes through public internet, all bets are off. I don't think all companies have their private clouds at the basement level, completely owned data centers, connected via edge switches near the water cooler.

Unless you uses Onion routing, yes

You're a shared tenant on a shared system. You are at the mercy of what other tenants do.

It could, but it currently doesn't. Regulating and enforcing censorship is a lot harder when the offending material is not publicly accessible / easy to find. So even if you have offending content on your own private cloud, as opposed to have someone elses offending content on a shared public cloud, the censors would first have to find out about it.