Distributed energy systems play a significant role in the integration of renewable energy technologies. The Energy Internet links a fleet of distributed energy systems to each other and with the grid. Interactions between the distributed energy systems via information sharing could significantly enhance the efficiency of their realtime operation. However, privacy and security concerns hinder such interactions. A game-theoretic approach can help in this regard, and enable consideration of some of these factors when maintaining interactions between energy systems. Although a game-theoretic approach is used to understand energy systems' operation, such complex interactions between the energy systems are not considered at the early design phase, leading to many practical problems, and often leading to suboptimal designs. The present study introduces a game-theoretic approach that enables consideration of complex interactions among energy systems at the early design phase. Three different architectures are considered in the study, i.e., energy eystem prior to grid (ESPG), fully cooperative (FCS), and non-cooperative (NCS) scenarios, in which each distributed energy system is taken as an agent. A novel distributed optimization algorithm is developed for both FCS and NCS. The study reveals that FCS and NCS reduce the cost, respectively, by 30% and 15% compared to ESPG. In addition to cost reduction, there is a significant change in the energy system design when moving from FCS to NCS scenarios, clearly indicating the requirement for a scenario that lies between NCS and FCS. This will lead to reducing design costs while maintaining privacy.