Acting as artificial synapses, two‐terminal memristive devices are considered fundamental building blocks for the realization of artificial neural networks. Current memristive crossbar architectures demonstrate the implementation of neuromorphic computing paradigms, although they are unable to emulate typical features of biological neural networks such as high connectivity, adaptability through reconnection and rewiring, and long‐range spatio‐temporal correlation. Herein, self‐organizing memristive random nanowire (NW) networks with functional connectivity able to display homo‐ and heterosynaptic plasticity is reported thanks to the mutual electrochemical interaction among memristive NWs and NW junctions. In particular, it is shown that rewiring and reweighting effects observed in single NWs and single NW junctions, respectively, are responsible for structural plasticity of the network under electrical stimulation. Such biologically inspired systems allow a low‐cost realization of neural networks that can learn and adapt when subjected to multiple external stimuli, emulating the experience‐dependent synaptic plasticity that shape the connectivity and functionalities of the nervous system that can be exploited for hardware implementation of unconventional computing paradigms.