Is it possible to put signatures (simple texts) to the Word documents before the user starts downloading the file?
You could do this with a custom solution to manipulate the OpenXML (assuming 2007+ format).
There is an older MSDN blog post that details the following:
You cant really control all the download methods, so you will want to capture it on upload to make your changes.
You might be able to utilize Information Rights Management to add labels to your Word documents when opened.