If you want to be completely safe, obviously you'd never post anything, you'd never visit these sorts of sites, you'd never have any of this content. Clearly we choose to disregard that. You should be using a VPN or TOR or something.
If you're an owner who doesn't share, and doesn't post content, the chances are you're not going to get caught unless you do something especially stupid.
If you fence hop, you run the risk of being caught.
I believe Blackstratu5 was sharing his dog with women and got caught that way. Real shame, considering he has the best mare videos out there.
You're not going to have your computer seized for investigation by the police unless you've been doing other stuff like drug dealing, CP, or anything else that will get you in serious trouble with the law. Or you decide to leave your zoo porn and let a tech poke around your PC and report it.
The short version is people get caught for taking risks. Producing porn, sharing animals, fencehopping.
As much as I'd love to get a chance if I did make friends with the right person, if I was an owner myself for security reasons I'd probably never meet up with anyone or produce any content. Thankfully others are a bit more relaxed there.