There is no gossip protocol that is built over Bitcoin Cash right now. The only way to send data over the network is to create a transaction.

If we want to use a gossip protocol, users will have to run federated servers and that requires a different incentive system. In this system, users pay a small fee and are assured that their message reaches the intended party.

New addresses can be used for new chats, since these addresses can be generated randomly, there is no way to connect them to an identity. Money can be routed to the address through a Bitcoin mixer to increase anonymity. (https://news.bitcoin.com/coingeek-to-fund-the-bch-anonymizer...)