A private data query system (PDQS) enables data consumers to query about privately-owned data by compensating the data owners for their loss of privacy. The two main tasks of a PDQS include procurement, i.e., collecting data from data owners with an appropriate pricing scheme, and querying, i.e., aggregating the collected dataset for a query output while preserving data owners’ privacy. Existing PDQS were designed to handle unconditional queries with a single attribute. In this paper, we design PrivCQ, a new PDQS for conditional queries over multi-dimensional data. For this, we introduce a novel privacy concept, multi-dimensional personalised local differential privacy (m-PLDP), to specify the privacy requirement over multiple sensitive attributes of different data owners. We propose total purchased privacy maximisation, a principle that bridges query accuracy with m-PLDP, and design two query mechanisms that achieve m-PDLP.