There is no adversarial process in getting the warrant, but the people on whom the warrant is served are given a document explaining the evidence that led to the warrant. The police give it to you when they do their search, or after the search is complete if it was something like a wiretap.
If the document does not support the need for a search or contains evidence that is fraudulently obtained, you can fight it either by striking the evidence at your trial or by separately suing the bad actor. At a minimum, during your trial, the warrant becomes part of the record and is publicly obtainable.
In the FISA case, the warrant document does not contain the evidence against you, since it is secret, barring you from fighting its validity later.
Also, when a crime is in progress, police don't need a warrant if they see it. They can just intervene. However, they can't search you for evidence of further crimes without probable cause of that other crime or a warrant.