One of the challenges of manufacturing hollow parts featuring internal lattice structures by using material extrusion is to achieve geometric accuracy of the internal geometries. In this work a solution for in-process geometric inspection and monitoring is presented, based on combining on-machine part measurement by laser line scanning and digital twinning. In the solution, a laser line scanner is used to acquire a two-dimensional map of material and void distribution within the deposited layer. Layer inspection is carried out by comparing the 2D map with a reference one obtained by simulating the deposition process (digital twin of the layer); discrepancies are automatically identified and quantified. The evolution of anomalies across layers can be tracked by vertically stacking both layer measurements and 2D digital twins and by investigating the resulting 3D voxel models. The models are updated after the fabrication of each new layer, to allow geometric monitoring over time. The proposed inspection and monitoring solution is particularly suitable for hollow parts and/or lattice or otherwise reticular internal structures, which would otherwise be inaccessible when using conventional measurement methods on the final part.