The fluid antenna system (FAS) has emerged as a disruptive technology, offering unprecedented degrees of freedom (DoF) for wireless communication systems. However, optimizing fluid antenna (FA) positions entails significant computational costs, especially when the number of FAs is large. To address this challenge, we introduce a decentralized baseband processing (DBP) architecture to FAS, which partitions the FA array into clusters and enables parallel processing. Based on the DBP architecture, we formulate a weighted sum rate (WSR) maximization problem through joint beamforming and FA position design for FA-assisted multiuser multiple-input multiple-output (MU-MIMO) systems. To solve the WSR maximization problem, we propose a novel decentralized block coordinate ascent (BCA)-based algorithm that leverages matrix fractional programming (FP) and majorization-minimization (MM) methods. The proposed decentralized algorithm achieves low computational, communication, and storage costs, thus unleashing the potential of the DBP architecture. Simulation results show that our proposed algorithm under the DBP architecture reduces computational time by over 70% compared to centralized architectures with negligible WSR performance loss.