Co Derry-born Patricia Devlin, an award-winning crime reporter with the Sunday World, took the case to the Police Ombudsman last year.
Her lawyer said her treatment by the PSNI was “deeply unsettling”.
The reporter had the support of Amnesty International.
Ms Devlin received the threat a year ago in a direct message to her Facebook account, signed in the name of neo-Nazi group Combat 18.
Police Ombudsman, Marie Anderson, said the threat made against the journalist was “repulsive”.
She added that it was “concerning that police failed to take measures to arrest the suspect at the earliest opportunity”.
A Police Ombudsman review of the investigation found that “evidential opportunities” were missed in regards to police enquiries.
A spokesperson said the police officer in question had been serving for six months at the time of the report.
The Ombudsman said that the officer “failed to take appropriate measures to secure the arrest of the suspect, who lived in another part of the UK”.
The Police Ombudsman recommended that the officer should be disciplined and said the PSNI accepted there had been failings and implemented measures to improve the officer’s performance.
Mrs Anderson said that “it is understandable that it caused distress and alarm”.
She added that she was considering “a broader policy recommendation” to the PSNI to ensure that officers across the service were “also made aware of the issues raised by this case”.
Ms Devlin’s lawyer, Kevin Winters, said questions have to be asked as to “why selective incompetence was allowed to facilitate the suspect in evading prosecution”.
“It is both deeply unsettling that a well-known journalist should be treated in this way by the PSNI and regrettably, on a wider level, it undermines confidence in policing in Northern Ireland,” he said.
Patrick Corrigan, of Amnesty International, said the threats sent to Patricia Devlin were “totally abhorrent” and are part of a “wider climate of intimidation of journalists in Northern Ireland”.
“It is the responsibility of the media to shine a light into the dark corners of our society,” he said.
“It is the police’s responsibility to uphold their right to do so and to ensure there is no impunity for those who threaten press freedom.
He added that the failure of the PSNI to “properly investigate this case is totally unacceptable”.
A PSNI spokesperson said: “A complaint was made to the Office of the Police Ombudsman, the case was upheld and the matter was dealt with under performance measures.”
Earlier this week Ms Devlin tweeted an update in her case, saying: “The PSNI informed me today (Tuesday) that after sending a file to the PPS on an individual I reported last year for online abuse, and who also contacted my place of work, will now be formally interviewed.
“It’s a small but positive step.”
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