The crosstalk between osteoblasts and endothelial cells is critical for bone vascularization and regeneration. Here, we used a coaxial 3D bioprinting method to directly print an osteon-like structure by depositing angiogenic and osteogenic bioinks from the core and shell regions of the coaxial nozzle, respectively. The bioinks were made up of gelatin, gelatin methacryloyl (GelMA), alginate, and hydroxyapatite (HAp) nanoparticles and were loaded with human umbilical vascular endothelial cells (HUVECs) and osteoblasts (MC3T3) in the core and shell regions, respectively. Conventional monoaxial 3D bioprinting was used as a control method, where the hydrogels, HAp nanoparticles, MC3T3 cells, and HUVECs were all mixed in one bioink and printed from the core nozzle. As a result, the bioprinted scaffolds were composed of cell-laden fibers with either a core-shell or homogenous structure, providing a non-contact (indirect) or contact (direct) co-culture of MC3T3 cells and HUVECs, respectively. Both structures supported the 3D culture of HUVECs and osteoblasts over a long period. The scaffolds also supported the expression of osteogenic and angiogenic factors. However, the gene expression was significantly higher for the core-shell structure than the homogeneous structure due to the well-defined distribution of osteoblasts and endothelial cells and the formation of vessel-like structures in the co-culture system. Our results indicated that the coaxial bioprinting technique, with the ability to create a non-contact co-culture of cells, can provide a more efficient bioprinting strategy for printing highly vascularized and bioactive bone structures.