Nowadays, sensor-equipped mobile devices allow us to detect basic daily activities accurately. However, the accuracy of the existing activity recognition methods decreases rapidly if the set of activities is extended and includes training routines, such as squats, jumps, or arm swings. Thus, this paper proposes a model of a personal area network with a smartphone (as a main node) and supporting sensor nodes that deliver additional data to increase activity-recognition accuracy. The introduced personal area sensor network takes advantage of the information from multiple sensor nodes attached to different parts of the human body. In this scheme, nodes process their sensor readings locally with the use of recurrent neural networks (RNNs) to categorize the activities. Then, the main node collects results from supporting sensor nodes and performs a final activity recognition run based on a weighted voting procedure. In order to save energy and extend the network's lifetime, sensor nodes report their local results only for specific types of recognized activity. The presented method was evaluated during experiments with sensor nodes attached to the waist, chest, leg, and arm. The results obtained for a set of eight activities show that the proposed approach achieves higher recognition accuracy when compared with the existing methods. Based on the experimental results, the optimal configuration of the sensor nodes was determined to maximize the activity-recognition accuracy and reduce the number of transmissions from supporting sensor nodes.