We theoretically study the transverse charge transport in Josephson junctions based on the tilted Dirac materials with valley-dependent gaps. It is shown that a finite tilt-assisted transverse Josephson Hall current is present under broken time-reversal symmetry. This transverse current is driven by the superconducting phase difference across the junction and exhibits a nonsinusoidal current-phase relation, leading to the transverse Josephson diode effect (TJDE), where the critical currents flowing oppositely along the transverse direction are asymmetric. Compared to the conventional longitudinal Josephson diode effect, the predicted TJDE supports a fully polarized diode efficiency with a 100% quality factor and can completely decouple the input signal path from the output, suggesting potential applications for nonreciprocal superconducting devices.