Considering groups of variables, rather than variables individually, can be beneficial for estimation accuracy if structural relationships between variables exist (e.g., spatial, hierarchical or related to the physics of the problem). Group-sparsity inducing estimators are typical examples that benefit from such type of prior knowledge. Building on this principle, we show that the diffusion LMS algorithm for distributed inference over networks can be extended to deal with structured criteria built upon groups of variables, leading to a flexible framework that can encode various structures in the parameters to estimate. We also propose an unsupervised online strategy to differentially promote or inhibit collaborations between nodes depending on the group of variables at hand.