No, but in some cases it does, I think it's when you comment on a manga in their own section. But mostly no.
Here's the description at the bottom of the "Want to read" page just before the tags.
What Manga will be shown here?
When you click ‘want to read/reading/have read’ button in manga page, this manga will be shown here. You can record and show manga here, it’s just like make your own manga gallery to show to yourself and other manga fans.
Now, when the manga listed in READING or READ list updated, you'll get notification!
There's also a description on reading and have read, you can check those out.

If I added a story to "want to read" library, will I be getting a notification if it gets updated?