Blockchain technology promises to overcome trust and privacy concerns inherent to centralized information sharing. However, current decentralized supply chain management systems do either not meet privacy and scalability requirements or require a trustworthy consortium, which is challenging for increasingly dynamic supply chains with constantly changing participants. In this paper, we propose CCChain, a scalable and privacy-aware supply chain management system that stores all information locally to give companies complete sovereignty over who accesses their data. Still, tamper protection of all data through a permissionless blockchain enables on-demand tracking and tracing of products as well as reliable information sharing while affording the detection of data inconsistencies. Our evaluation confirms that CCChain offers superior scalability in comparison to alternatives while also enabling near real-time tracking and tracing for many, less complex products.