In this article, we concentrate on a generic multiterminal joint source-channel coding scenario, appearing in a wide variety of real-world applications. Specifically, several noisy observations from a source signal must be compressed at some intermediate nodes, before getting forwarded over multiple error-prone and rate-limited channels towards a (remote) processing unit. The imperfections of the forward channels should be integrated into the design of (local) compressor units. By following the Information Bottleneck principle, the Mutual Information is selected here as the fidelity criterion, and a novel (data-driven) design approach is presented for two distinct types of processing flow/strategy at the remote unit. To that end, tractable objective functions are developed, together with the pertinent learning architectures, generalizing the concepts of Variational Auto-Encoders and (Distributed) Deep Variational Information Bottleneck for (remote) source coding to the context of distributed joint source-channel coding. Unlike the conventional approaches, the proposed schemes here work based upon a finite sample set, thereby obviating the call for full prior knowledge of the joint statistics of input signals. The effectiveness of these novel sample-based compression schemes is substantiated as well by a couple of simulations over typical transmission setups.