Here's how to do it.
This is a screenshot of the Home stream on Google+.
You can start a post using the pencil icon or clicking in the "What's new with you?" space.
Clicking either will open the post editor.
Attention! The post editor by default has the same audience setting as the last post you composed. In this screenshot the post I made previously was to my Me me me, personal posts Collection.
We are wanting to make a private post, so we'll need to change the "audience" setting.
Click the audience link to open the audience selector.
Tip: I always change the audience FIRST. If you compose your post first it's easy to hit post having forgotten to change the audience!
What you see here will depend on the Collections you have set up. If you have numerous collections you will need to click the "See More" search area to see more of your available choices and also the search option.
Click "search" to activate the search option.
Start typing the name of the person you want the post to be visible to in the search area and choose the name when it appears in auto-complete.
Note: you can repeat the process to make the post visible to more than one person if you need to.
After you have chosen the audience click "done" to return to the post editor.
You'll see that the audience "to" area now reflects who will be able to see your post.
Note: that although your post will be visible to the people or audience you have selected they control whether they will receive a notification about the post or not.
I have my notification settings open i.e. set to public, so people can reach me. I also have the App installed on my phone. Obviously if people do not have either then they might not see your post, but that's their fault, not yours.
Finish up & post!
Add any text, media, photos, poll, location you want to include before sending. You also have other post options like turning off resharing or closing comments, but they can also be change after the post has been made.
Here is the process in GIF form.
If you are making private posts often I really recommend using a special collection for them, as it shortens the process and keeps all your private messages in one place. Learn about that here!