The neural computations underlying sensory-guided behaviors can best be understood in view of the sensory stimuli to be processed under natural conditions. This input is often actively shaped by the movements of the animal and its sensory receptors. Little is known about natural sensory scene statistics taking into account the concomitant movement of sensory receptors in freely moving animals. South American weakly electric fish use a self-generated quasi-sinusoidal electric field for electrolocation and electrocommunication. Thousands of cutaneous electroreceptors detect changes in the transdermal potential (TDP) as the fish interact with conspecifics and the environment. Despite substantial knowledge about the circuitry and physiology of the electrosensory system, the statistical properties of the electrosensory input evoked by natural swimming movements have never been measured directly. Using underwater wireless telemetry, we recorded the TDP of Apteronotus leptorhynchus as they swam freely by themselves and during interaction with a conspecific. Swimming movements caused low-frequency TDP amplitude modulations (AMs). Interacting with a conspecific caused additional AMs around the difference frequency of their electric fields, with the amplitude of the AMs (envelope) varying at low frequencies due to mutual movements. Both AMs and envelopes showed a power-law relationship with frequency, indicating spectral scale invariance. Combining a computational model of the electric field with video tracking of movements, we show that specific swimming patterns cause characteristic spatiotemporal sensory input correlations that contain information that may be used by the brain to guide behavior.