One would think that being an admin and using Discourse on multiple sites for a few years I would know this, but so untrue.
As noted in the original post I thought the use of @ would invite someone to a private/personal message. But that is not the case.
Also as noted there is the invite button but after searching the entire topic and checking all the buttons including the wrench it turns out there is no button named invite. Instead just under the first post is
It is pretty obvious how it works so I will not elaborate.
The reason for adding the reply here is because this topic is the first result Google offered.