The electronic transport and the sensing performance of an individual SnO2 crossed-nanowires device in a three-terminal field-effect transistor configuration were investigated using a combination of macroscopic transport measurements and scanning surface-potential microscopy (SSPM). The structure of the device was determined using both scanning electron- and atomic force microscopy data. The SSPM images of two crossed one-dimensional nanostructures, simulating a prototypical nanowire network sensors, exhibit large dc potential drops at the crossed-wire junction and at the contacts, identifying them as the primary electroactive elements in the circuit. The gas sensitivity of this device was comparable to those of sensors formed by individual homogeneous nanostructures of similar dimensions. Under ambient conditions, the dc transport measurements were found to be strongly affected by field-induced surface charges on the nanostructure and the gate oxide. These charges result in a memory effect in transport measurements and charge dynamics which are visualized by SSPM. Finally, scanning probe microscopy is used to measure the current-voltage characteristics of individual active circuit elements, paving the way to a detailed understanding of chemical functionality at the level of an individual electroactive element in an individual nanowire.