The critical brain hypothesis proposes that neural networks operate near criticality to reap the computational benefits of accessing a wide range of timescales [1]. Proponents of this viewpoint highlight the presence of heavy-tailed (power-law) spatiotemporal correlations as markers of criticality in the brain [2]. Critics of this hypothesis argue that such correlations could be inherited from upstream sources, such as sensory input [3]. Similarly, Ref. [4] constructed a model of independent neurons driven by shared noise input that exhibited neural activity with power-law tails, which could be interpreted as introducing an infinite correlation range inherited by otherwise independent neurons. Determining whether the brain is critical thus demands a way to distinguish intrinsically generated criticality from heavy-tailed input correlations inherited from upstream input. We derive a mean-field theory to investigate the related effects of intrinsic criticality and heavy-tailed inputs using a model of spiking neural activity driven by external noise (Fig. 1.A) in which the spiking network and the noise process can be tuned independently to a critical state (Fig. 1.A-B). In the networks we consider these critical states correspond to the boundary at which a single steady firing rate state becomes unstable to self-sustaining low- or high-activity states. We show that the autocovariance of spiking activity is heavy-tailed when the input is critical, irrespective of the degree of criticality in the network (Fig. 1.C, main panels). Conversely, the response functions of the network---measured as the network response to a current impulse averaged across trials---are only heavy-tailed if the spiking network is tuned to criticality (Fig. 1.C, insets). These causal responses of neurons to membrane perturbation are independent of the input, rendering the criticality of the input irrelevant. Our work thus suggests experimentally observed response functions can disambiguate intrinsic versus inherited criticality in spiking neural networks.