Efficient compression of correlated data is essential to minimize communication overload in multi-sensor networks. In such networks, each sensor independently compresses the data and transmits them to a central node. A decoder at the central node decompresses and passes the data to a pre-trained machine learning-based task model to generate the final output. Due to limited communication bandwidth, it is important for the compressor to learn only the features that are relevant to the task. Additionally, the final performance depends heavily on the total available bandwidth. In practice, it is common to encounter varying availability in bandwidth. Since higher bandwidth results in better performance, it is essential for the compressor to dynamically take advantage of the maximum available bandwidth at any instant. In this work, we propose a novel distributed compression framework composed of independent encoders and a joint decoder, which we call neural distributed principal component analysis (NDPCA). NDPCA flexibly compresses data from multiple sources to any available bandwidth with a single model, reducing compute and storage overhead. NDPCA achieves this by learning low-rank task representations and efficiently distributing bandwidth among sensors, thus providing a graceful trade-off between performance and bandwidth. Experiments show that NDPCA improves the success rate of multi-view robotic arm manipulation by 9% and the accuracy of object detection tasks on satellite imagery by 14% compared to an autoencoder with uniform bandwidth allocation.