Introducing noise in the training of machine learning systems is a powerful way to protect individual privacy via differential privacy guarantees, but comes at a cost to utility. This work looks at whether the inherent randomness of stochastic gradient descent (SGD) could contribute to privacy, effectively reducing the amount of \emph{additional} noise required to achieve a given privacy guarantee. We conduct a large-scale empirical study to examine this question. Training a grid of over 120,000 models across four datasets (tabular and images) on convex and non-convex objectives, we demonstrate that the random seed has a larger impact on model weights than any individual training example. We test the distribution over weights induced by the seed, finding that the simple convex case can be modelled with a multivariate Gaussian posterior, while neural networks exhibit multi-modal and non-Gaussian weight distributions. By casting convex SGD as a Gaussian mechanism, we then estimate an `intrinsic' data-dependent $\epsilon_i(\mathcal{D})$, finding values as low as 6.3, dropping to 1.9 using empirical estimates. We use a membership inference attack to estimate $\epsilon$ for non-convex SGD and demonstrate that hiding the random seed from the adversary results in a statistically significant reduction in attack performance, corresponding to a reduction in the effective $\epsilon$. These results provide empirical evidence that SGD exhibits appreciable variability relative to its dataset sensitivity, and this `intrinsic noise' has the potential to be leveraged to improve the utility of privacy-preserving machine learning.