Carbon nanotube field effect transistors (CNTFETs) are of great interest for nanoelectronics applications such as nonvolatile memory elements (NVMEs) or charge sensors. In this work, we use a scanning-probe approach based on a local charge perturbation of CNTFET-based NVMEs and investigate their fundamental operation from combined transport, electrostatic scanning probe techniques and atomistic simulations. We experimentally demonstrate operating devices with threshold voltages shifts opposite to conventional gating and with almost unchanged hysteresis. The former effect is quantitatively understood as the emission of a delocalized image charge pattern in the nanotube environment, in response to local charge storage, while the latter effect points out the dominant dipolar nature of hysteresis in CNTFETs. We propose a simple model for charge sensing using CNTFETs, based on the redistribution of the nanotube image charges. This model could be extended to gas or biosensing, for example.