Part of being trusted with power is the ability to do things without transparency - to be able to filter what and what not you share. Done correctly, this will increase the trust in you in both public and private matters.
I agree with that in the interpersonal realm. There are downsides to over-sharing. But maybe it was confusing for me to frame it as trusting. It's more like permission to be investigated. Or the absence of penalties for investigation. That is, reporters and police have the right (and indeed the duty) to investigate the powerful.