In my opinion, the news report is mischaracterizing the situation, either because the person who provided the information was confused or sloppy, or because the reporter was confused (it is impossible to tell on its face).
Probable cause to arrest can, and often does, arise from an oral statement of a witness to a police officer, even if a witness refuses to back their oral statement to a police officer with a written one.
And, in the narrow and strict sense, a complaint is a court document, usually signed by a prosecutor, commencing criminal legal proceedings against a criminal defendant (in lieu of an indictment, or pending the issuance of an indictment). A complaint in this narrow and strict sense would not usually be signed by an ordinary citizen witness to a crime.
Sometimes, however, the word "complaint" is used in a broader, non-technical sense to refer either to any report made to police complaining about misconduct by someone, or in a different technical sense to refer to a written and signed report made by a witness to the police.
It is likely that what really happened is that after receiving an oral report that the individual had threatened people, none of the witnesses who provided this oral report was willing to sign a document summarizing their oral report of being threatened for the police (perhaps out of fear of retaliation by the accused person, or out of a desire not to harm the accused person's long term future prospects).
Then, the police department, as a matter of department policy (rather than any requirement imposed by law), declined to pursue a criminal case based upon the threats, when no one was willing to publicly and in writing commit to their oral reports, because they would need a testifying witness in a later court case.