The convergence of Opportunistic Networks and Trusted Execution Environments at the network edge presents a compelling opportunity for fully decentralized privacy-preserving data processing. Based on this convergence, we define the concept of Edgelet computing, a new paradigm for executing powerful and privacy-preserving distributed queries on personal devices. Our objective is to establish a robust, secure, and scalable execution framework with strong individual privacy guarantees. This paper first proposes a liability model tailored to decentralized executions on crowd members' devices, along with a query evaluation model that differs from the traditional database closed-world assumption. Second, it defines essential properties for ensuring the security, resiliency and validity of executions, and subsequently presents several methods and strategies for their enforcement. Through a comprehensive qualitative analysis and extensive evaluations, we showcase the relevance and effectiveness of the approach, demonstrating that Edgelet Computing holds potential for the emergence of novel and important classes of applications.