The completion, extension, and generation of 3D semantic scenes are an interrelated set of capabilities that are useful for robotic navigation and exploration. Existing approaches seek to decouple these problems and solve them one-off. Additionally, these approaches are often domain-specific, requiring separate models for different data distributions, e.g.\ indoor vs.\ outdoor scenes. To unify these techniques and provide cross-domain compatibility, we develop a single framework that can perform scene completion, extension, and generation in both indoor and outdoor scenes, which we term Octree Latent Semantic Diffusion. Our approach operates directly on an efficient dual octree graph latent representation: a hierarchical, sparse, and memory-efficient occupancy structure. This technique disentangles synthesis into two stages: (i) structure diffusion, which predicts binary split signals to construct a coarse occupancy octree, and (ii) latent semantic diffusion, which generates semantic embeddings decoded by a graph VAE into voxel-level semantic labels. To perform semantic scene completion or extension, our model leverages inference-time latent inpainting, or outpainting respectively. These inference-time methods use partial LiDAR scans or maps to condition generation, without the need for retraining or finetuning. We demonstrate high-quality structure, coherent semantics, and robust completion from single LiDAR scans, as well as zero-shot generalization to out-of-distribution LiDAR data. These results indicate that completion-through-generation in a dual octree graph latent space is a practical and scalable alternative to regression-based pipelines for real-world robotic perception tasks.