Aren't we also free to trade amongst ourselves during a market circuit breaker event?

Yes, but it's much, much slower to trade directly than through a market-maker in the exchange

how does one actually technically go about that? I have no plans to do it, just been curious about it over the past few months.

In my part of the world, tell your broker that you’ve done a deal with someone, tell that someone to tell their broker that they done a deal with you. The brokers should be able to report it to the exchange for publishing. Then they should see through the clearing and settlement. Of course your brokers might feel that your little deal is just not worth the hassle, and ask you to take your business elsewhere.

You ask your broker for the physical copy of the stock certificate. The buyer asks their bank for some physical cash. They hand you the cash, you hand them the stock certificates. Same as buying or selling anything else, basically.

Of course, for a transaction of a certain value, you wouldn't want to hold the cash or certificate, so instead you write out a contract and sign it at the moment of exchange. Which is the same as buying or selling something else (e.g. car, house) beyond a certain value.