Even in P2P, you would still have to pay through a bank account or something. Or any e-wallet that most likely have your identity but at least you have more freedom in the method you can use to pay and there is only one person that you have to transact with as opposed to an intermediary you do not even know consists of who.
In P2P trades, your trade partner knows your personal information like your full name and your bank account information. The P2P marketplace has those information in user chat log too. They surely have access to user chat information, for user disputes and other things. This means, your privacy is lost after you publish in online, even in P2P trades and in user chat for a trade.
Since we are a community here who understand privacy, I am sure there is nothing to worry about here.
It's wrong belief. It's your individual issue, your privacy and you can not rely on community to defend your privacy. There are not all members in the forum community care about privacy.