Smart home applications have increasingly adopted compute-intensive, machine-learning based techniques to improve their efficiency and prediction capabilities, e.g., for the comfort of the residents. While edge computing has been the preferred computing paradigm in smart homes due to the latency, data privacy, and resilience benefits, the limited computational and bandwidth resources of local edge devices, e.g., a home gateway, might hinder the timely completion of Internet-of-Things (IoT) tasks, necessitating computation offloading of some tasks to remote compute nodes. While ensuring timely completion of an application is essential in offloading decisions, offloading without consideration of data privacy poses privacy risks. To mitigate such consequences, this paper presents RASH, a resource allocation and offloading scheme that has three merits. First, different from prior studies, RASH assigns a privacy-sensitivity score to the tasks based on their data type and the location of the IoT devices to prioritize local execution of highly sensitive tasks when local resources are insufficient and offloading is inevitable. Second, to handle peak load scenarios, RASH incorporates a heuristic task postponement algorithm that effectively defers tasks to maintain the problem’s feasibility and maximize resource utilization. Third, RASH runs periodically, dynamically adjusting resource allocation based on real-time computational demands and newly arriving training and executing tasks with varying time budgets and computational needs. Our evaluation results show that RASH effectively prioritizes the local execution of privacy-sensitive tasks with more than 90% local resource utilization.