you can use filters http://docs.xp-dev.com/user-guide/filters.html to see all completed bugs, though there’s no notification per-se just for completed bugs or when the filter gets updated. Having said that, it does seem like an interesting feature.

On the other hand, you can configure your notifications from http://www.xp-dev.com/settings/subscriptions That will give you the ability to control notifications just for you.