There is a lot of misunderstanding about what Private or “Incognito” mode really means and that can get people into trouble.
Basically those safe modes mean that the browser won’t inadvertently store any information about your online activity during the private session. History won’t be saved, cookies will be automatically deleted at the end of the session (when the tab or window is closed, etc.). If someone were to sit down at the machine and try and figure out what you had been up to… these private modes should make that very difficult.
But incognito or safe mode (the two terms are used interchangeably here) won’t hide intentional activity. If you intentionally download a file that file will stay on your computer. If you intentionally create a bookmark that bookmark will remain. This would make it much easier for someone to sit at your machine and figure out what you had been up to.
And incognito mode won’t affect anything outside of your browser. The ISPs and networks between you and the sites you visit, and those sites themselves, do not care about or respect your browser mode. They are still going to track you as much as they can.
This is especially true if you log in to a website. If for example you log in to your LinkedIn account while in private mode and visit profiles… those visits won’t appear in the history of your browser, but they will appear in your LinkedIn activity.
ISPs will also be tracking you, so don’t think private mode means you can illegally download files without consequences. Authorities and copyright holders will still be able to trace your connection.
Gizmodo has more great information about the reality of private browsing modes.