In networks of neurons, correlations and synaptic strengths are heavy-tailed, with a small number of neurons interacting much more strongly than the vast majority of pairs. Yet, it remains unclear whether, and how, such heavy-tailed connectivity emerges from simple underlying mechanisms. Building upon recent advances in neuroimaging, here we show that the correlations between thousands of neurons in the mouse visual cortex have heavy-tailed distributions spanning three decades in strength. Moreover, we find that these distributions are robust to variations in the visual stimuli, even persisting during spontaneous activity. To explain these heavy-tailed correlations, we propose a simple model of synaptic self-organiazation based on Hebbian plasticity: synapses are pruned at random, and the synaptic weight is redistributed throughout the network in either (i) a Hebbian fashion or (ii) randomly. Importantly, our model contains only a single parameter $0 \le p \le 1$, which represents the probability of Hebbian versus random growth. We predict analytically and confirm numerically that such dynamics generate scale-free distributions of connectivity strength, with a power-law exponent $\gamma = 1 + \frac{1}{p}$ that depends only on the probability of Hebbian growth. Finally, by generalizing our model to artificial neural networks, we demonstrate that Hebbian plasticity gives rise to heavy-tailed correlations similar to those observed in neuronal recordings. Generally, our results suggest that heavy-tailed distributions of correlations and synaptic weights may arise from general principles of self-organization, rather than the biophysical particulars of individual neural systems.