I agree that this could be useful, and I think the best place for it is the inbox, since it's a notification about something you signed up for. (You could probably make an argument for it being in the achievements box because not every proposal makes it into private beta, so it is something of an achievement, but I think the inbox is more intuitive.)
It could look something like this:
With the following as a template for the notification text:
private beta invitation
{Site Name}
This proposal has entered private beta; click here to join and start asking and answering!
Update: starting with the next launch (as of June 9, 2015), private beta notifications will be sent to people who committed to a proposal or followed it during the definition phase.
