- There has been a growing interest in non-Hermitian physics [1]. One of its main goals is to engineer dissipation and to explore ensuing functionality. Since magnons are intrinsically damped in magnets, its dynamics aligns well with the goal of non-Hermitian physics [2]. Although the effect of dissipation due to local damping on magnon transport has been explored, the effects of non-local damping on the magnonic analog of the Josephson effect [3] remain missing, despite that non-local damping is inevitable in magnets. Here, we uncover theoretically that rich dynamics can emerge in magnetic junctions due to intrinsic non-local damping, using analytical and numerical methods [4]. In particular, under microwave pumping, we show that coherent spin precession in the right and left insulating ferromagnet (FM) of the junction becomes synchronized by non-local damping and thereby a magnonic analog of the $\varphi$ Josephson junction emerges, where $\varphi$ stands here for the relative precession phase of right and left FM in the stationary limit. Remarkably, $\varphi$ decreases monotonically from $ \pi$ to $\pi/2$ as the magnon-magnon interaction increases. Moreover, we also find a magnonic Josephson diode effect giving rise to rectification of magnon currents. Our predictions are readily testable with current device and measurement technologies.