There is no in-built functionality for this in the current forum software. You could suggest for it to be added to Epochtalk if you wanted:
https://bitcointalk.org/index.php?board=167.0If you want to achieve this now, you will need to do so using a third party tool.
https://visualping.io/ is probably the most commonly used. Paste in the URL of someone's "Latest posts" or "Latest topics" page, and it will email you whenever that page changes i.e. they make a new post/topic.