The ever-growing computational demands of increasingly complex machine learning models frequently necessitate the use of powerful cloud-based infrastructure for their training. Binary neural networks are known to be promising candidates for on-device inference due to their extreme compute and memory savings over higher-precision alternatives. In this paper, we demonstrate that they are also strongly robust to gradient quantization, thereby making the training of modern models on the edge a practical reality. We introduce a low-cost binary neural network training strategy exhibiting sizable memory footprint reductions and energy savings vs Courbariaux & Bengio's standard approach. Against the latter, we see coincident memory requirement and energy consumption drops of 2--6x, while reaching similar test accuracy in comparable time, across a range of small-scale models trained to classify popular datasets. We also showcase ImageNet training of ResNetE-18, achieving a 3.12x memory reduction over the aforementioned standard. Such savings will allow for unnecessary cloud offloading to be avoided, reducing latency, increasing energy efficiency and safeguarding privacy.