There is a duty in the legislation for Multi-Academy Trusts to maintain the SCR, covering all Schools within the Trust. This means that the Trust should hold a centrally maintained record of the checks that have been carried out across all Schools.
The Trust can maintain that information, either on one single document or on a separate document for each School. In practice, it is likely that the School would hold a copy of the centrally held SCR, that relates to its own employees. Therefore, where the School is responsible for carrying out the checks that must be recorded on the SCR (such as the identity check), the School can inform the Trust to ensure the centrally maintained record is updated.
Regardless of how the centrally held SCR is maintained it should allow for a particular School's information to be easily extracted so that it can be provided to an inspector or other person who is eligible to see the information and avoid disclosing details of employees who do not work at that particular School. In many Schools, the SCR is managed by a member of the business team, typically a senior manager.
Although there is no statutory guidance on this, ultimately, the Headteacher and the Chair of Governors are responsible for the correct vetting of employees.