Multi-Access Edge computing represents one of the most important enabling technologies for the Industrial Internet of Things. It allows advanced data processing and customized service provisioning, very close to the end-users. In the presence of many Multi-Access Edge computing applications, however, it is fundamental to ensure effective and privacy-preserving data dissemination at the network edge. From the security perspective, Attribute-based Encryption and Searchable Encryption techniques can be jointly used to achieve data confidentiality, flexible protection against unauthorized access, and privacy-preserving data dissemination. Available solutions, however, generally focus the attention on cloud-based approaches, use edge computing to implement some of the cryptographic tasks, and limit the investigation to single cryptographic operations. Indeed, no works investigate the adoption of these techniques in scenarios with multiple data producers and end-users, and fully operating at the network edge. To bridge this gap, this work proposes a novel methodology supporting fast and privacy-oriented data dissemination directly at the network edge. In the considered distributed network infrastructure, Multi-Access Edge computing applications express the interest to receive specific data by sending Trapdoors to Edge Servers. Data sources protect their contents through Attribute-based Encryption and deliver them to Edge Servers. In turn, Edge Servers implement Attribute-based Searchable Encryption functionalities to properly disseminate received contents towards Multi-Access Edge nodes hosting the applications that generated valid Trapdoors. The performance of the conceived approach has been evaluated through computer simulations. Obtained results highlight the benefits achieved against baseline (i.e., cloud-based) solutions.