There is something else to consider here. Coinbase bends over backwards for regulators and law enforcement. They even do blockchain analysis on where you send your coins. People have had their accounts terminated for sending BTC from their Coinbase accounts to gambling sites. I don't even want to know what they do if you're sending BTC to a darknet market.
As far as what Coinbase shares with the site receiving the payment -- not much. They can see that the coins are coming from Coinbase and the IP address of the sender. Like I said, that would not be my first concern.
Not purchasing anything on darknet market..don't even know what it is or how I would, still trying to figure out bitcoin, LOL! It's an adult site that I am trying to remain private on.
You should be fine, then. As far as I know, Coinbase has no issues with adult entertainment. But it's generally good practice to move coins to your own wallet when spending from Coinbase, to avoid a provable connection between you and an illicit site (or a gambling site). I'd hate to get my Coinbase account closed over something like that.
But yeah, they aren't going to give your personal information to any merchant. The merchant can only see the coin source (Coinbase's hot wallet) and their node's IP address. Nothing about you.