A private profile is a user who does not want anyone following them without their consent.
To be invisible on Saverd, simply mark your profile as private, don't follow anyone or accept any follow requests and set your recipes to "Only me" in the visibility section.
If you want to interact with selected people you can accept the follow requests of those people and only those people will see things you post. (Based on recipe visibilty settings of course).
To request to follow someone simply search for them and select their profile. From there the only thing you will see (since the user is private) will be the request follow button:
Click the request button which will result in the following:
If anyone requests to follow you there will be a review follow requests button under your profile in the left hand column and it will look like this:
Clicking on the above button will list your follow requests in the center feed where you can accept them or ignore them.