The power system requires an additional amount of flexibility to process the large-scale integration of renewable energy sources. Community Energy Storage (CES) is one of the solutions to offer flexibility. In this paper two scenarios of CES ownership are proposed. Firstly, an Energy Arbitrage (EA) scenario is studied where an aggregator aims to minimize costs and CO2-emissions of an energy portfolio. Secondly, an Energy Arbitrage-Peak Shaving (EA-PS) scenario is assessed, which is based on a shared ownership between a Distribution System Operator (DSO) and an aggregator. A multi-objective Mixed Integer Linear Programming (MILP) optimization model is developed to minimize the operation costs and CO2-emissions of a community situated in Cernier (Switzerland), using different battery technologies in the CES system. The results demonstrate a profitable system design for all Lithium-ion-Batteries (LiBs) and the Vanadium Redox Flow Battery (VRFB), for both the EA and EA-PS scenarios. The economic and environmental performance of the EA-PS scenario is slightly worse compared to the EA scenario, due to power boundaries on grid absorption and injection to achieve peak shaving. Overall, the differences between the EA and EA-PS scenarios, in economic and environmental performance, are small. Therefore, the EA-PS is recommended to prevent problematic loads on the distribution transformer. In addition, the Pareto frontiers demonstrate that LiBs perform best on both economic and environmental performance, with the best economic and environmental performance for the Lithium-Nickel-Manganese-Cobalt (NMC-C) battery.