On the encouragement of Kuba, I will collect in an answer the multiple comments (with due credit to rm -rf, Jagra, and Daniel Lichtblau) that do, to a large extent, answer the question. For me, the most important points are the following:
Mathematica notebooks do not, by themselves, store explicit personally identifying information such as author names, as you might get with e.g. MS Word, and pdfs printed from it.
However, there can be certain types of information in the notebook (including but not limited to cell editing times, custom style sheet locations, and the file path or filename of the notebook) that can show up in the notebook or pdfs printed from it, and which can reveal traits (such as timezone / active times, operating system, or the very use of Mathematica itself) which, depending on context, can end up identifying you.
To remove the cell editing times, go to Cell $\to$ Notebook History and click Clear History. To remove the file outline cache, go to Format $\to$ Options Inspector, choose Show Option Values $\to$ Selected Notebook, go to Notebook Options $\to$ PrivateNotebookOptions, and change
"FileOutlineCache" to False. Doing this will make the notebook much easier to read in a file editor, and therefore help in finding any further information you might wish weren't there.
Most importantly, notebooks can be opened externally with any text editor; with the exception of Graphics objects they are quite transparent to read. Thus, if this is a concern, you can and should have a good look at the notebook to see if there's anything there that you'd rather not be.
For the examples above, disabling CellChangeTimes will remove cell editing times from the notebook; turning off FileOutlineCache in the options inspector will leave only the notebook expressions behind, though paths to the stylesheets will remain. If you print a pdf from the notebook, it may still be a good idea to run it through a pdf metadata removal tool such as this one if metadata is an issue, just to be sure. The notebooks themselves, though, will have nothing in them that's not visible through a text editor.