Simultaneous recordings from the cortex have revealed that neural activity is highly variable and that some variability is shared across neurons in a population. Further experimental work has demonstrated that the shared component of a neuronal population's variability is typically comparable to or larger than its private component. Meanwhile, an abundance of theoretical work has assessed the impact that shared variability has on a population code. For example, shared input noise is understood to have a detrimental impact on a neural population's coding fidelity. However, other contributions to variability, such as common noise, can also play a role in shaping correlated variability. We present a network of linear-nonlinear neurons in which we introduce a common noise input to model—for instance, variability resulting from upstream action potentials that are irrelevant to the task at hand. We show that by applying a heterogeneous set of synaptic weights to the neural inputs carrying the common noise, the network can improve its coding ability as measured by both Fisher information and Shannon mutual information, even in cases where this results in amplification of the common noise. With a broad and heterogeneous distribution of synaptic weights, a population of neurons can remove the harmful effects imposed by afferents that are uninformative about a stimulus. We demonstrate that some nonlinear networks benefit from weight diversification up to a certain population size, above which the drawbacks from amplified noise dominate over the benefits of diversification. We further characterize these benefits in terms of the relative strength of shared and private variability sources. Finally, we studied the asymptotic behavior of the mutual information and Fisher information analytically in our various networks as a function of population size. We find some surprising qualitative changes in the asymptotic behavior as we make seemingly minor changes in the synaptic weight distributions.