Model compression has emerged as an important area of research for deploying deep learning models on Internet-of-Things (IoT). However, for extremely memory-constrained scenarios, even the compressed models cannot fit within the memory of a single device and, as a result, must be distributed across multiple devices. This leads to a distributed inference paradigm in which memory and communication costs represent a major bottleneck. Yet, existing model compression techniques are not communication-aware. Therefore, we propose Network of Neural Networks (NoNN), a new distributed IoT learning paradigm that compresses a large pretrained 'teacher' deep network into several disjoint and highly-compressed 'student' modules, without loss of accuracy. Moreover, we propose a network science-based knowledge partitioning algorithm for the teacher model, and then train individual students on the resulting disjoint partitions. Extensive experimentation on five image classification datasets, for user-defined memory/performance budgets, show that NoNN achieves higher accuracy than several baselines and similar accuracy as the teacher model, while using minimal communication among students. Finally, as a case study, we deploy the proposed model for CIFAR-10 dataset on edge devices and demonstrate significant improvements in memory footprint (up to 24x), performance (up to 12x), and energy per node (up to 14x) compared to the large teacher model. We further show that for distributed inference on multiple edge devices, our proposed NoNN model results in up to 33x reduction in total latency w.r.t. a state-of-the-art model compression baseline.