The increasing integration of Distributed Energy Resources (DERs) within power distribution grids introduces uncertainties that can result in control and operation issues such as line congestions, reduced voltage quality and increased grid imbalance cost. Existing research tackles these issues through topology reconfiguration or reinforcement of existing assets, such as lines and transformers, as well as allocation of new assets. However, these strategies may lead to an inefficient allocation of financial and human resources, since they solve specific optimization problems without a holistic view of the infrastructure and stakeholders involved. In this respect, this paper presents a stochastic two-stage local market-aware Energy Storage Systems (ESSs) allocation model which aims at optimally siting and sizing ESSs within a fair local market environment, thereby enhancing the effective activation of local resource flexibility. First, price fairness is defined in terms of quality of experience (QoE), taking into account the characteristics of the local distribution network. Under the proposed price fairness condition, the site and size of ESSs are determined by a planning stage. With this optimal allocation of ESSs, the operation stage makes use of a local market cleared based on the Augmented Relaxed Optimal Power Flow (AROPF) and primal-dual method to maximize social welfare. To solve the resulting Mixed-Integer Second-Order Cone Programming (MISOCP) problem, the Benders decomposition approach is applied. Case studies conducted on the IEEE 33-bus system illustrate that the proposed market-aware ESSs allocation model improves effectiveness of investment utilization, local flexible resources activation and local market fairness.