This research discusses fairness in energy communities while investigating two types of organizations for operation and cost-sharing. On the one hand, centralized architectures consist of operating community assets in a coordinated manner with a central controller before a community manager shares the overall benefits between the users. Four sharing strategies are investigated and implemented in a monthly post-delivery phase. In contrast, in decentralized architectures, each user operates its assets independently. In such frameworks, the costs/benefits are usually shared among users through market-based mechanisms that rely on users' bids. This work then explores the Pool market and Peer-to-peer transactions to investigate the impact of different bidding from the users' perspective. Ultimately, all the proposed centralized and decentralized approaches (10 in total) are assessed based on economic performances at both users' and community levels. Specific attention is paid to fairness within the community, which is challenging. Three indexes derived from economy and game theory are then considered, along with metrics tailored for energy communities. Results from a seven-user community indicate that the pool market systematically returns considerable savings among decentralized frameworks compared to peer-to-peer markets. More importantly, centralized frameworks systematically yield the most significant bill reduction (16 %) and fairer cost allocation compared to decentralized framework.