If you want to keep streams everywhere and avoid allocating huge arrays of bytes, which is good practise (for example, if you plan to post big files), you still can do it with a derived version of WebClient. Here is a sample code that does it.

using (var client = new WebClientWithResponse())
{
    using (var stream = client.OpenWrite(myUrl))
    {
        // open a huge local file and send it
        using (var file = new FileStream(filePath, FileMode.Open, FileAccess.Read, FileShare.ReadWrite))
        {
            file.CopyTo(stream);
        }
    }

    // get response as an array of bytes. You'll need some encoding to convert to string, etc.
    var bytes = client.Response;
}

And here is the customized WebClient:

public class WebClientWithResponse : WebClient
{
    // we will store the response here. We could store it elsewhere if needed.
    // This presumes the response is not a huge array...
    public byte[] Response { get; private set; }

    protected override WebResponse GetWebResponse(WebRequest request)
    {
        var response = base.GetWebResponse(request);
        var httpResponse = response as HttpWebResponse;
        if (httpResponse != null)
        {
            using (var stream = httpResponse.GetResponseStream())
            {
                using (var ms = new MemoryStream())
                {
                    stream.CopyTo(ms);
                    Response = ms.ToArray();
                }
            }
        }
        return response;
    }
}