Hi @TanguyLauc 
All transaction data relevant for a particular party is stored on the participant node that serves it/where it’s located. In that sense all transactions are private as only data relevant for a party that is involved in a transaction is stored on the node.
Having said that there are some general trust assumptions between different services running on a node as @Ratko_Veprek points out here. Going over the example topology from Canton’s Getting Started guide provides more info as well as the basic example of how transactions are executed in Canton