Just go to the store page of a game that has a demo, and tap/click “Download” where it says download demo.
You mention “the library is shared between Linux and windows”, which is concerning.
If you are mounting an NTFS windows partition inside Linux so you don’t have to redownload your games, that won’t work.
Running games installed onto an NTFS partition is really fiddly on Linux, and steam WILL break things when you try to run games installed on linux in windows, and vice versa.