Shadar-Kai primarily speak Elvish, though Sylvan is also commonly spoken, with their particular accent being described as harsh and guttural. Common is almost non-existent in day-to-day activities and is typically only learned by elders and envoys to communicate with outsiders.. Most things there would either not speak or would speak whatever they knew before going there. Things like ghosts or vampires would speak whatever they spoke when they were alive. Other living things would speak languages based on where they came from as well. You might consider them speaking undercommon which is associated with the underdark but is kind of similar to the Plane of Shadow as well.
Shadar-kai were an offshoot of elves bound to the Shadowfell after the Raven Queen whom they served fell.[1] Shadar-kai in the Shadowfell looked like withered, almost corpse-like elves, with wrinkled skin and swollen joints; on other planes they appeared more youthful. Their hair was pale and their skin had a deathly pallor no matter the location. They preferred to wear cloaks and veils.[1.. DND 5th Edition