Local Electricity Markets (LEMs) arise as new layers of current market designs, enabling local trading of flexibility products. In many cases, LEMs encompass several networks, involving different Distribution System Operators (DSOs). This setting raises intrinsic concerns regarding the information privacy of the involved DSOs while trying to achieve system-wide efficiency. In this paper, a market design for the trading of flexibility products in inter-DSO LEMs is proposed where the DSOs are allowed to trade among the areas under the coordination of a Local Market Operator (LMO). A coordinated and decentralized approach based on the Alternating Direction Multiplier Method is proposed. The novel aspect of this work is that each DSO selfschedules its own assets in response to market signals (optimal dual variables) to fulfil the flexibility requests from itself or from other DSOs. An illustrative case study based on the IEEE 123 bus test systems is used for testing the proposed framework.