They don't exist exactly as stated in your question.
There are badges -- Announcer, Booster and Publicist -- for sharing a link to a post using the
share button underneath the post. (More information in the Q&A: How can I 'try' to get the publicist badge?)
On Area 51, you can get both reputation for each person that signs up for a site proposal via the "Share this" link on the proposal's page, and badges -- Lobbyist, Campaigner and Activist -- if enough people sign up using your referral link.
But on this site, there are no badges for referring people that eventually join the site, despite several feature requests on Meta Stack Exchange, for example:
Given that the blog post about the Announcer, etc. badges is presented as an answer for some of the above Meta SE feature requests, presumably the hope is that people that visit using the referral link will join the site and become productive members.