The statistics of fluctuations in biological sensing pathways and its relation to the response to environmental stimuli is investigated. We focus on bacterial chemotaxis, where detailed experiments and reliable models are available. We consider allosteric models of receptors' activity and derive analytically their steady-state probability distribution and correlation times. By using fluctuation relations, we then relate appropriate steady-state correlations to the response of the system to step and ramp stimuli of arbitrary amplitudes. We show that the combined effect of nonlinearity and fluctuations generically yields a complex nonlinear response at the single sensing unit and at the whole-cell level. Such responses display a nonexponential decay with a broad range of time scales. Slow, ineffective responses are associated to signaling units locked into poorly performing states. However, the nonlinear response reduces to a nearly exponential one for an appropriate range of the kinetic parameters. This provides a systematic explanation for the relation between fluctuation and response observed in recent experiments.