We can have PowerShell eat an .exe's STDOUT very easily:

xcopy file.txt target.txt | Out-Null;

We can have it eat both STDOUT and STDERR by redirecting STDERR into STDOUT

xcopy file.txt target.txt 2>&1 | Out-Null;
xcopy "ThisFileDoesn'tExist.ps1" target.txt 2>&1 | Out-Null;

Unix's filehandles are somewhat clever.  You can redirect STDOUT to a file, then redirect STDERR into STDOUT, and eat STDOUT.  This nukes STDERR.  I'll leave the advisablity of this to the user.

cp Schroedinger.txt target.txt> found.txt 2>&1 > dev/null

However, this doesn't work in PowerShell:

xcopy Schroedinger.txt target.txt | Set-Content found.txt 2>&1 | Out-Null

If worst comes to worst, we can use cmd.exe to do this.

cmd /c xcopy Schroedinger.txt target.txt "2>NUL"

Ugh.  Anyone have any better ideas?