We present a numerical investigation of collective resonances in lattices of dielectric nanoparticles. These resonances emerge from the enhanced radiative coupling of localized Mie resonances in the individual nanoparticles. We distinguish two similar systems: a lattice of silicon nanoparticles homogeneously embedded in a dielectric and a lattice of silicon nanoparticles in an optical waveguide. The radiative coupling is provided by diffraction orders in the plane of the array for the former system or by guided modes in the optical waveguide for the latter one. The different coupling leads to distinct lattice resonances in the metasurface defined by the array of silicon nanoparticles. These resonances have been extensively investigated in metallic nanoparticle arrays, but remain highly unexplored in fully dielectric systems. We describe the pronounced differences in the intensity enhancement and field distributions for the two systems, providing valuable information for the design and optimization of optical components based on dielectric lattice resonances.