Let's say that your buddy, Pete, comes over to your house. He parks his car in your driveway, knocks on the door, and you let him in. There's a cop parked in his car across the street, and he's watching all this. But if he wants to come into your house against your will, he has to get a warrant, and have probable cause. Let's examine this scenario.
Your house is located in the public.
Pete drove his car in the public, and onto your private driveway.
The cop is in the public, and watches from the public.
All kinds of things related to you and Pete are in the public eye.
What you and Pete do and transact behind closed doors in private. Unless there is probable cause and a warrant, the cop isn't supposed to break into your privacy.
Now, lets look at Bitcoin.
Bitcoin is public.
The code is open source = public.
The blockchain is public.
The public addresses of each of you are public.
The NSA might even be able to track that it is you and Pete who did such and such a transfer.
But the transaction is private until you or Pete disclose the reasons for it.
The private addresses that you and Pete each have automatically inside your Btc client are private to each of you.
Everything is both public and private. It is this way for all people and governments and nations. Privacy in the US is strong until people want to give it up. Foundational US law upholds the privacy, in the 4th Amendment, and in loads of Supreme Court decisions going way back to 1800.
It is only when Bitcoin exchanges open themselves up to the public that they make themselves vulnerable to public laws that government might pass. And if people use these exchanges, they lose some of their privacy, even though the details of the reason behind any transaction remains private.
The point is to make some of the exchanges to be private. This means that they can't advertise in public ways. It all has to be word of mouth to get anybody to join.
An example of this is the failed
Paraiba fraudulent securities company -
https://outranklab.com/is-paraiba-world-a-scam. The leaders and heads of this Ponzi got away with it because they didn't advertise publicly. Everything was done by word of mouth, and an applicant had to be accepted privately. Info leaked out, of course, but it never leaked from the official company, except by word of mouth, person to person.
Paraiba was a scam, a Ponzi. But we know that Bitcoin is legit. The point is, an exchange that is advertised only by word of mouth among its members can be private. This means, no exchange website. Being private takes it outside the rules of public law. Examining how
Paraiba was set up and 'advertised', might be the best way to set up a private exchange, outside of government control.
Note that, for all we know, there might be many little private exchanges... exchanges that do not advertise, but grow by word of mouth among members and potential members. We don't know about them, because nobody talks about them publicly.