The separation principle in a centralized estimation and control problem gives us the flexibility to design a feedback controller independent of the state estimator. However, the same principle does not hold when the estimation and control are distributed over a network of agents. In this case, the estimator may need to be redesigned when the controller is revised, which can be computationally expensive. We investigate a weaker notion of the separation principle in the distributed estimation and control of linear time-invariant (LTI) systems. As a main contribution, applying the small-gain theorem, we characterize the notion using matrix inequalities and compute a set of feedback controllers that agents in the network can adopt without redesigning the estimator. We also analyze how the frequency of information exchange between neighboring agents affects the characterization. We illustrate our analytical results through simulations of a multi-vehicle system problem.