Demand-side energy management (EM) is studied from a privacy-cost trade-off perspective, considering time-of-use pricing and the presence of an energy storage unit. Privacy is measured as the variation of the power withdrawn from the grid from a fixed target value. Assuming non-causal knowledge of the household’s aggregate power demand profile and the electricity prices at the energy management unit (EMU), the privacy-cost trade-off is formulated as a convex optimization problem, and a low-complexity backward water-filling algorithm is proposed to compute the optimal EM policy. The problem is studied also in the online setting assuming that the power demand profile is known to the EMU only causally, and the optimal EM policy is obtained numerically through dynamic programming (DP). Due to the high computational cost of DP, a low-complexity heuristic EM policy with a performance close to the optimal online solution is also proposed, exploiting the water-filling algorithm obtained in the offline setting. As an alternative, information theoretic leakage rate is also evaluated, and shown to follow a similar trend as the load variance, which supports the validity of the load variance as a measure of privacy. Finally, the privacy-cost trade-off, and the impact of the size of the storage unit on this trade-off are studied through numerical simulations using real smart meter data in both the offline and online settings.