I'd like to know if I'm leaking personal data if I hop from computer to computer running my
.tex file without deleting some hidden log of some kind that retains, say, an address I have in an invoice within my TeX doc. Am I safe?
Out of the box LaTeX stores information about the used program version and the used packages in the
.log file. In other auxiliary files it stores certain information about your document such as section header information for the table of contents, cross reference information, etc. basically anytihng that LaTeX needs to know about the document as a whole and therefore stores it externally in one processing run to reuse it in the next run.
Such data may contain PII (personal identifiable information) depending on what you put into, say, section headers. The content of the
\author command used in
\maketitle is not transfered to such external files (by default).
However, as it was mentioned in comments, LaTeX is a full general programming language that can write file so in principle any data in your source document may be processed and externalized in a file. So depending on additinal packages you use other PII data may show up in auxiliary files.
So if you want to be safe, delete all helper files after processing your documents as they are not needed and can be recreated. By default they are all written in the local directory. But again, nothing prevents a malicious package to write to some other place in the file system (in theory). In practice you should be safe assuming you use a standard distribution.