The global shift towards decentralised energy systems has assigned municipalities a key role in achieving national climate neutrality objectives. As the main stakeholders in the local energy transition, municipalities are responsible for the decarbonization of the local energy system through the extensive integration of renewable energy sources into existing systems. However, this integration requires new approaches and system adjustments, such as energy storage deployment, to satisfy the variable nature of renewable energy sources. The integration of novel solutions, such as energy storage, is difficult because of the diverse range of stakeholders involved, each with their own perceptions and expertise. This study uses the Fuzzy Cognitive Mapping (FCM) methodology to analyse the mental models of different stakeholders regarding their perceived importance of different factors influencing the implementation of energy storage in municipalities. The approach of this study enables a better understanding of municipal energy systems and its dynamics. The results reveal that support schemes such as subsidies and awareness campaigns are key to all stakeholders. Municipalities tend to focus on local needs and technological solutions, while energy experts prioritize technical aspects and national policies. Municipalities address challenges linearly, missing interconnections, whereas energy experts consider feedback loops and system requirements. The study highlights the need for common ground to drive effective policy and infrastructure development. The results could be used to facilitate discussions with policy makers on why energy storage is important and what policy measures should be considered to accelerate its deployment.