It depends mainly on your ISP and whether or not different organizations are monitoring the torrent you want to download. Larger ISPs like AT&T, Charter, and Comcast can monitor customer traffic. Small local companies, like what I am using, don't usually.
Public torrents are easy to access, therefore more likely to be watched for pirating. When you start downloading your IP gets recorded. Organizations can then notify the police or your ISP. Private torrent sites, where you have to request an invite to join and then have to monitor your download/upload ratio to maintain good standing, are much easier to continue downloading without getting caught.
I've never gotten caught downloading anything. I've downloaded a ton of stuff, too. I download more now that I am not using AT&T and am using a local company.