Co-clustering exploits the duality of instances and features to simultaneously uncover meaningful groups in both dimensions, often outperforming traditional clustering in high-dimensional or sparse data settings. Although recent deep learning approaches successfully integrate feature learning and cluster assignment, they remain susceptible to noise and can suffer from posterior collapse within standard autoencoders. In this paper, we present the first fully variational Co-clustering framework that directly learns row and column clusters in the latent space, leveraging a doubly reparameterized ELBO to improve gradient signal-to-noise separation. Our unsupervised model integrates a Variational Deep Embedding with a Gaussian Mixture Model (GMM) prior for both instances and features, providing a built-in clustering mechanism that naturally aligns latent modes with row and column clusters. Furthermore, our regularized end-to-end noise learning Compositional ELBO architecture jointly reconstructs the data while regularizing against noise through the KL divergence, thus gracefully handling corrupted or missing inputs in a single training pipeline. To counteract posterior collapse, we introduce a scale modification that increases the encoder's latent means only in the reconstruction pathway, preserving richer latent representations without inflating the KL term. Finally, a mutual information-based cross-loss ensures coherent co-clustering of rows and columns. Empirical results on diverse real-world datasets from multiple modalities, numerical, textual, and image-based, demonstrate that our method not only preserves the advantages of prior Co-clustering approaches but also exceeds them in accuracy and robustness, particularly in high-dimensional or noisy settings.