Mycelium Local Trader uses your bitcoin private key for registration and authentication with Mycelium servers. The only information retained by Mycelium servers is your bitcoin address, nickname, sell orders with the entered location, and your trade history.
To enhance privacy, Local Trader uses a coarse location based on your network address, or an approximation near the address you manually enter. Your location is only updated when you manually change it from the Change location menu.
Local Trader will log all actions that involve the server, along with the time and the trader nickname. These logs are used for debugging purposes, and deleted after a few days. All trades are of course permanently stored in the blockchain.
All chat conversations are end-to-end encrypted using your bitcoin keys, and their context is only accessible to the people involved in the chat. The only things Mycelium servers receive is encrypted text with time stamps.