James Allen was jailed for life in November 2012 for killing Colin Dunford, 81, in Middlesbrough, and Julie Davison, 50, in Whitby.
An investigation commissioned by NHS England found there were "missed opportunities" in Allen's case.
A spokesman said it was clear his care fell "well below expected standards".
Blackpool-born Allen's trial heard he had numerous previous convictions, and had served an eight-year jail sentence for causing grievous bodily harm.
The independent investigation was carried out by Niche Patient Safety to see if lessons could be learned by those involved in his care.
At the time of the offences, Allen was receiving care from Tees, Esk and Wear Valleys NHS Foundation Trust, provider of mental health services.
The report found there was enough evidence to indicate Allen was a "vulnerable individual" and services should have identified that there was a "significant probability" that he would reoffend.
However, what was "not predictable" was his choice of victims.
Allen attacked and killed Mr Dunford at his home on 23 April 2012 while lying low at a friend's house after being accused of a serious crime.
Ms Davison was discovered dead in her flat two days later.
The report, which refers to Allen as Mr F, concluded: "Mr F was a serial offender who was either unwilling or unable to engage in any meaningful rehabilitation programme.
"The evidence indicates that there were many deficiencies and missed opportunities by both primary and secondary health care services where important information could have been sought and shared.
"If obtained, this information would have enabled a more accurate assessment of Mr F's risk factors and would have alerted agencies to his potential for reoffending."
Karen Conway, head of investigations for NHS England North, said: "The circumstances surrounding the tragic death of these two victims are extremely upsetting and our deepest sympathies go to their family and friends.
"The report rightly highlights a number of issues around roles, responsibilities and working practices which must be resolved for positive progress to be made... we will be ensuring that all recommendations relating to primary care are implemented and embedded into general practice."
A man who brutally murdered two people while on the run from police was "vulnerable" and had "significant known risk factors", a report has found.