The spreading of misfolded proteins underlies a wide range of neurodegenerative diseases known as proteinopathies. Recent mathematical models based on partial differential equations (PDEs) have enabled detailed studies of misfolding processes. Numerical discretization of these models poses challenges due to their nonlinearities and the complex boundaries and interfaces of brain geometry. Polytopal numerical methods, leveraging mesh agglomeration of fine tetrahedral meshes, effectively address this complexity. Various models describe prion spreading at different levels of detail, such as the heterodimer model or the Fisher--Kolmogorov equation. We introduce and analyze polytopal discontinuous Galerkin methods for the semi-discrete approximation of these models, with a focus on accurately simulating the wavefronts observed in prion propagation. A key challenge is preserving the entropy structure and positivity of solutions, which prevents instabilities from unstable equilibrium points. To this end, we develop structure-preserving numerical methods by reformulating the problem with an entropy variable in a local discontinuous Galerkin (LDG) framework. Finally, we simulate alpha-synuclein spreading in a two-dimensional sagittal brain slice using a polygonal agglomerated grid. Our results reproduce typical patterns observed in Parkinson's disease and dementia with Lewy bodies.