Yes they can but only if they use a third party tools like Orangedox or a link shortener like Bit.ly or Google URL shortener. Though for all of those the URL will be different than the default https://drive.google.com/open?id=xxxxx

You can see the number of people in real time who are viewing your Google Docs, simply open the doc and in the right hand corner of the screen you'll see the number of unique users who are currently viewing the document. In the screenshot below you’ll see that there are three unique users viewing the document.

So unless the person who shared the doc with you has it open at the same time they won't know that you've viewed or downloaded it.

If someone did want to know when you've viewed or downloaded their document I'd consider moving your document to Google Drive and using a document tracking plugin like Orangedox | Track your Documents. This will let you track every preview of your document including each unique user who’ve viewed or downloaded it.

Disclaimer: I work for Orangedox