Private Computing on Set Intersection (PCSI) addresses the limitations of Private Set Intersection (PSI) by enabling computation over intersection while preserving its privacy. However, more real-world applications require the parties to perform calculations on the intersection and its labels, which the traditional PCSI does not support. In this paper, we design an innovative PCSI framework for computing arbitrary functions of the intersection and their labels. The core of the PCSI framework is our newly proposed Secret Labeled PSI (SLPSI) protocol, which outputs homomorphic cipher labels of the intersection to support arbitrary follow-up computations. To achieve efficient SLPSI, particularly in unbalanced settings, we present a Scalable Dual-Load Bloom Filter (SDLBF), which achieves a false positive rate of approximately \(0.4\%\) , an initialization time of around 20 s, and a query time of about 40 \(\mu \) s—metrics that are noteworthy in their own right. Leveraging SDLBF, our SLPSI protocol accomplishes an average \(65\%\) reduction in online communication cost compared to the leading protocol by Chen et al. (CCS 2018). We further demonstrate our PCSI framework’s superiority in a real-world privacy-preserving inference system, showcasing improved practicability and utility over existing solutions.