Police are not giving schools enough information about prospective employees who pose a risk to pupil safety, headteachers said this week.
But teacher unions said their members needed protection against false and malicious allegations being passed on to schools.
The Secondary Heads Association said in its submission to the Bichard Inquiry into the Soham murders that details of investigations, as well as convictions, should be disclosed. This would enable schools to make more informed judgments when taking on new staff.
In its submission, the National Association of Schoolmasters Union of Women Teachers said there should be an "appropriate balance between protecting children and the rights of individuals".
The inquiry into the murders of Holly Wells and Jessica Chapman will investigate how Ian Huntley got a job as a school caretaker despite a series of previous sex allegations against him.
It is being carried out by Sir Michael Bichard, the former permanent secretary at the Department for Education and Skills.
During the trial it emerged that Humberside police destroyed important evidence which linked Huntley to sex offences against young girls, believing they had to do so under the Data Protection Act.
Cambridgeshire police, meanwhile, failed to send on all of his personal details for checking.
SHA and the NASUWT both called for consistent practice between police forces.
SHA said information should be shared with schools for child protection reasons.
"We recommend that this should include details of any convictions and investigations that, on the balance of probability, suggest a risk may exist," its submission said.
It went on: "Headteachers cannot act effectively to fulfil their duty of care because the information on which they have to base decisions is inconsistent and sometimes inadequate."
The NASUWT said awareness should be raised of teachers' vulnerability to false and malicious allegations, which should be reflected in the way information is recorded and disclosed.
Three weeks of public hearings will begin on February 16.