> that would be implying that "private" really means anything for AWS

Insert your dedicated hosting provider of choice for 'AWS' (somewhere like Hetzner will be cheaper anyway).

But in general, AWS hosts are yours, running your code, with your security policies enforced. Sure, the US government can silently subpoena the contents thereof, but aside from that fairly extreme case, it's not like AWS is handing your data over to 3rd parties.