First of all, anonymity and privacy are two different things. Bitcoin is designed to be pseudo-simple – which comes close to anonymity, but instead of being completely anonymous, you are linked to your Bitcoin address or pseudonym.
You can not make a private transaction on the Bitcoin network, but you can try to be as anonymous as possible.
It is possible to run Bitcoin Core as a hidden Tor service and connect to such services.
The first step is to run Bitcoin Core behind a Tor proxy. This will make all outgoing connections anonymous, but more is possible.
Further instructions at Github: