Spell check was made available in Telegram Desktop as of version 1.9.3 and, while it worked fine with Ubuntu-based distros, it was not working on Arch-based ones. It didn’t take me too long to figure it out but I wasn’t sure if others had yet.
If you are having this issue, make sure you install the aspell dictionary for your language. In my case, it was aspell-en. Restart Telegram and it should work.
I made a quick video showing how.