In normal usage the degree to which something is either public or private is called "privacy". That's the term used by websites to describe the settings that control who can see or modify content. It's also used for other similar attributes before websites.
I'll meet you at the Serpentine Bridge.
How much privacy will we get there?
Not much, it's very public.
Then I'd like to go somewhere with more privacy please
or
We have lots of privacy here. Let's go skinny-dipping.
or
Everyone in the rooms around us can hear us. We don't have much privacy.
Also you are wrong (in general) about "public" and "private" being dichotomous opposites. It may be true in the system you are writing, but it hardly ever is in real life. In English things can be "somewhat private", "very public", etc. Almost all programming languages have several levels of privacy ("public", "private" and "protected" are very common). And websites have different level of privacy - Anyone, Friends, Close Friends, Friends of Friends etc. Having only two values of "public" and "private" is extremely unusual.
You could also use "visibility" if it was specifically about whether something can be seen, or "accessibility" if it's about whether something can be interacted with.
protected
andinternal
are two that come to mind from C#.