The debug log can be started on both the client and the server, but they act independently of each other.
Starting the debug log on a client saves information regarding events in a debug log file on the client machine and the information saved is specific to that client. Starting the debug log from a client does not affect the server or any other clients. It only creates a debug log for that specific client and saves the event information for that specific client.
Starting the debug log on the Server saves information regarding events in a debug log file on the server machine and the information saved is specific to the server. Starting the debug log on the server does not affect any of the clients. It only creates a debug log and save information for the server and saves the event information for the server.