The emergence of decentralized social media platforms presents new opportunities and challenges for real-time analysis of public discourse. This study introduces CognitiveSky, an open-source and scalable framework designed for sentiment, emotion, and narrative analysis on Bluesky, a federated Twitter or X.com alternative. By ingesting data through Bluesky's Application Programming Interface (API), CognitiveSky applies transformer-based models to annotate large-scale user-generated content and produces structured and analyzable outputs. These summaries drive a dynamic dashboard that visualizes evolving patterns in emotion, activity, and conversation topics. Built entirely on free-tier infrastructure, CognitiveSky achieves both low operational cost and high accessibility. While demonstrated here for monitoring mental health discourse, its modular design enables applications across domains such as disinformation detection, crisis response, and civic sentiment analysis. By bridging large language models with decentralized networks, CognitiveSky offers a transparent, extensible tool for computational social science in an era of shifting digital ecosystems.