I suggest you use Bitso24 services (I am its programmer). Cheap prices start at $ 2 a month.
In these services, torrent is downloaded on the server and you will download it with a direct link.
So there will be no logs about you.
Public torrents are deleted immediately after the download is complete, but private torrents are different.
Private torrents are usually uploaded after the download is complete, which varies from site to site.
For example, in bitso24, private torrents are uploaded up to rate 6.