But there is no mechanism to share it explicitly with people you vetted. This isn't Facebook, there aren't friends with various levels of permission and a public version.
You could do that by just showing your screen to your vetted people, though, I suppose.
(a) There are friends with various levels of permission. Certain fields on the profile can be set to be visible only to certain groups of contacts.
(b) Upon noting (a), I filled out a private profile in anticipation that Google would be rolling out further mechanisms to share private profiles with certain contact groups.
You could do that by just showing your screen to your vetted people, though, I suppose.