Articles are determiners. Determiners' purpose in a sentence is to indicate "which X" the speaker/writer is talking about, for singular nouns.
The rules are simple but abstract and don't really depend on what X is. But they can depend on how the writer/speaker views things.
If you expect the listener/reader to know which X you are talking about, you say the X. Some reasons that you would expect the listener/reader to know which X you are talking about already is that you've just mentioned it, that it's a shared experience, or that there is only one X in the world.
Otherwise you use a/an if you aren't using any other determiner (e.g. it doesn't matter "which X" you are talking about.)
1 To search (a/the) private ownership (a/the) policeman must have a warrant.
- Do we expect the listener/reader to know which private ownership? If not, say "a private ownership." If "private ownership" refers to a private ownership you were talking about earlier, or that you expect the reader/listener to know what you mean, the private ownership should be expressed.
Example where you would say "the":
Emily owns a house, which is a type of private ownership. She was asking what the police need to search. To search the private ownership, a warrant is needed.
We say "the private ownership" because if we were asked "which private ownership", we would answer with something earlier in the sentence, that the speaker/listener is expecting us to remember or know already.
We do not use the article in the first occurence of "private ownership" because think about it - what's the answer to the question "which private ownership?" in that first sentence. None. We're talking about the type of something. No articles used when we are doing that. There is no actual instance of "private ownership" in that first sentence, until we tell the listener/reader that "Emily's house" is an instance of "private ownership."
We can do this too:
Emily owns a house, which is a private ownership. She was asking what the police need to search. They need a warrant to search the private ownership.
The subtle difference between this example and the previous is that we are saying "it doesn't matter which private ownership" in the above example. This implies there are other private ownerships nearby (the first example above does not imply it). After that first sentence is completed, we know we're talking abouy Emily's private ownership, which becomes "the" private ownership.