Personal and sensitive data is now collected at larger scales than ever before. Growing concerns of data subjects and regulatory bodies, however, have led to an increased demand for statistical procedures that do not compromise the privacy of the individuals whose data are collected and analysed. In this talk I will discuss recent work in the field of differential privacy, specifically local differential privacy (LDP), which aims to formalise notions of privacy and data security. Motivated by the difficulty of high-dimensional problems under LDP constraints, and by common settings in which each individual holds multiple data points, I will present results on the new user-level variant of LDP. These distributed problems involve interesting phase transitions and, for high-dimensional problems with sparsity constraints in particular, striking differences compared to behaviour in the standard model of LDP.