I think the screams might be words - from Words of Radiance page 672 (kindle first edition) when Kaladin performs the "last clap" on Relis' Shardblade during Adolin's duel (emphasis mine):
"Relis dropped the blade as if bitten. He backed away, raising his hands to his head. 'What is it? What is it! No, I didn't kill you!" He shrieked as if in great pain [...]"
Why would Relis say that if he only heard wordless screaming? My conjecture is the spren is screaming "YOU KILLED ME!" or something like that.
CaptainRyan's post in STORMLIGHT SPOILERSThe Silence of the Shardblades was marked as the answer