When you make us fetch a torrent for you, we download that torrent with our own torrent client which is in Netherlands. If someone is collecting IP addresses, then they see our IP address. And when you're streaming or downloading the fetched files from us, as far as your ISP is concerned, you're downloading a certain amount of data from a domain called put.io.
There are no public links, no one will see your files unless you become friends with them on put.io and share with them.
About our logging policies (or what if we get hacked or raided and they have our servers):
We log events globally without user information attached to them. We keep a log of every torrent transfer in order to troubleshoot problems: when it was started, how long it was seeded, when it was written to storage etc. But there is no userid in those logs.
BUT: There is an association with your account and the downloaded torrent, as long as it is on your Transfers page, and there is an association between your account and the files in that torrent, as long as they are visible among Your Files.
Also two more places that keep a record of what's going on are your History page and Trash (if enabled). You can clear your history by clicking Remove All near the subheader, and clear your your trash by clicking on Empty Trash. You can also disable Trash and History altogether from Settings.
Other than these, we keep no history. We wouldn't be able to answer the question "What has this user downloaded?" after you clear your files, transfers, trash, and history.
Also, if you ever delete you account, we destroy all traces of the account and its associated files, etc. The only record we keep is a log entry that states that an account with this certain username has deleted their account on a certain date. This is so that we can trace account deletions if it wasn't intended. Nothing else about your account stays in our systems.
Hope this answers any concerns you might have. Don't hesitate to contact us if you have further questions.