.NET Framework
Scrivi e leggi dallo stream di StdErr

Scrivi su output errore standard utilizzando Console

var sourceFileName = "NonExistingFile";
try
{
    System.IO.File.Copy(sourceFileName, "DestinationFile");
}
catch (Exception e)
{
    var stdErr = Console.Error;
    stdErr.WriteLine($"Failed to copy '{sourceFileName}': {e.Message}");
}

Leggi da errore standard del processo figlio

var errors = new System.Text.StringBuilder();
var process = new Process
{
    StartInfo = new ProcessStartInfo
    {
        RedirectStandardError = true,
        FileName = "xcopy.exe",
        Arguments = "\"NonExistingFile\" \"DestinationFile\"",
        UseShellExecute = false
    },
            
};
process.ErrorDataReceived += (s, e) => errors.AppendLine(e.Data);
process.Start();
process.BeginErrorReadLine();
process.WaitForExit();

if (errors.Length > 0) // something went wrong
    System.Console.Error.WriteLine($"Child process error: \r\n {errors}");