Point clouds produced by 3D sensors are often sparse and noisy, posing challenges for tasks requiring dense and high-fidelity 3D representations. Prior work has explored both implicit feature-based upsampling and distance-function learning to address this, but often at the expense of geometric interpretability or robustness to input sparsity. To overcome these limitations, we propose PU-Gaussian, a novel upsampling network that models the local neighborhood around each point using anisotropic 3D Gaussian distributions. These Gaussians capture the underlying geometric structure, allowing us to perform upsampling explicitly in the local geometric domain by direct point sampling. The sampling process generates a dense, but coarse, point cloud. A subsequent refinement network adjusts the coarse output to produce a more uniform distribution and sharper edges. We perform extensive testing on the PU1K and PUGAN datasets, demonstrating that PU-Gaussian achieves state-of-the-art performance. We make code and model weights publicly available at https://github.com/mvg-inatech/PU-Gaussian.git.