of the agent, varies from agent to agent. More concretely, the observation model we adopt in this paper is of the form, [$$\begin{aligned}
            \mathbf{y}_{n}(t) = \mathbf{H}_{n}\btheta^{\ast}+\mathbf{\gamma}_{n}(t),
            \end{aligned}$$]{} where $\btheta^{\ast}\in\mathbb{R}^{N}$, and the dimension of $\btheta^{\ast}$ corresponds to the number of physical locations being monitored in the field, $\mathbf{H}_{n}$ is a (fat) matrix that abstracts the coupling in measurements for agent $n$ with the state values at other nodes and $\mathbf{\gamma}_{n}(t)$ represents the observation noise (to be specified later). Existing distributed estimation schemes, such as in [@kar2011convergence; @kar2013consensus+; @Mesbahi-parameter; @Giannakis-est; @Sayed-LMS; @Stankovic-parameter; @Giannakis-LMS; @Nedic-parameter; @sahu2016distributed], aim to reconstruct the entire parameter at each node of the networked setup, thus reflecting a homogeneous objective across all nodes. However, in this paper, we consider a distributed estimation scheme which allows agents to pursue a heterogeneous objectives, in which agents’ only estimate a few components of the parameter vector $\btheta^{\ast}$ corresponding to their *interest sets*. Accounting for heterogeneity is highly relevant in practice. The heterogeneous objectives across agents lets us extend the notion of *consensus* to *subspace consensus*, where the agents reach consensus with respect to entries of the parameter that lie in the intersection of their interest sets. The second level of heterogeneity in 