You can still use the "Invite Fellow Experts" option (it's actually a bug, but you may use it as long as they don't fix it). You will find it here. Scroll down a bit. It should be on the right side of your screen if you're using "Desktop mode" for browsing. Just type in their email id and add a custom message (optional) and click on "send invite". Stack Exchange will send them a mail with a link to this site's main page (on your behalf i.e. your user id/name will be visible on the mail). The only downside is that it still shows "private" instead of "public".
Just in case you're curious, here's how a custom mail from Stack Exchange looks (when using the Invite Fellow Experts option):
The other options are, of course, directly inviting people you know, by providing them the link: https://quantumcomputing.stackexchange.com/ via mail, social media or word-of-mouth.
I'm not aware of any reward(s) associated with inviting people (in the public beta phase) to the site (even they sign up and/or are someone famous in the field). Anyhow, if you do want such badges to be introduced, better raise the issue on Meta Stack Exchange, as Stack Exchange is unlikely to introduce such badges for a single site only. However, do keep in mind that there are some inherent problems with "badges for being able to successfully persuade famous people to join the site". As I mentioned earlier, it is pretty much impossible to verify a person's real identity, online (at least without making some major changes to the Stack Exchange framework).