Yes, this is a confusing issue.
The primary use of the definite article is to distinguish a particular instance (of which the reader already knows or will come to know) from the generic type. For example "the house" vs "a house."
The terms public and private can be used as adjectives. Then preceding them by "the" is not appropriate. For example "public services" and "private property".
The public can be used as a noun, in which case it is always preceded by "the" and might also have an adjective. For example : "the general public."
Those are the simple cases.
More ambiguously, the public and the private can be used (with the definite article) to refer to abstract ideas or mutually exclusive "realms" in the political/economic arena. However, as FumbleFingers points out, in this case the meaning is much the same if we use public and private without "the" as adjectives. The difference - if there is one - is subtle.
Your examples demonstrate the abstract use, and work both ways with substantially the same meaning. The following possibilities are all correct (I have taken the liberty of changing "division") :
- The distinction between the public and the private is not very clear.
- The distinction between (what is) public and (what is) private is not very clear.
- The distinction between public (property) and private (property) is not very clear.
- The distinction between the public (sector) and the private (sector) is not very clear.
The comments by FumbleFingers are worth repeating :
The [definite] article is optional in both cases : changing public and private from nouns into adjectives modifying "space" doesn't significantly affect things.
If the speaker wishes to give the impression that the public [world] and the private [world] are two well-known realms, the case for the definite article is stronger - but of course it's weakened by a context where the speaker is claiming that we don't even know them well enough to make a clear distinction between them!