I have access to the server logs from my web hosts ( DotEasy.com), but that only gives me very general information about traffic volumes and search patterns, not identifiable to any person. If you would like to take a look then go to http://west-penwith.org.uk/webalizer/web/ (I won't put a clickable link here because that would attract the search engines with possible unfortunate results.)
That just about covers it.
All these suppliers may keep logs which contain other information but you will have to read their own privacy statements to find out if it is a problem (unfortunately, they are not very helpful).
There is a separate statement about this on the Introduction to the Database page.
I would be interested to hear from you if you have any problems or concerns with any of this.