I think anything set to private will probably not have huge SEO benefits because it will probably not be crawled and cached by search engines hence unlikely to carry any significant SEO benefits if any at all.
As others pointed out, you can choose to share Google+ posts with certain Circles. If you wish to maintain control over distribution, you need to make sure that you have disabled resharing. As for potential SEO benefits, anything shared privately will not appear in the search engines, and as such, I guess it has little or - more likley - no value. I'm not entirely sure whether +1s on Google+ posts shared privately will affect the rankings indirectly, i.e. when there is a link in the Google+ post that was +1d by someone: will that benefit the page that is linked?