Private schools - which receive funding or support from State levels - and/or are recognized by the State as an approved school, are under the same scrutiny for teacher certification as a public school. Teaching certification standards can vary from state to state. Certification has to be renewed every so many years as determined by the governing state. Teacher's are required to take continuing education courses, and meet certain advanced degree requirements to maintain certification. Beginning teacher's are required to have a bachelor's degree - and depending on they area they are certified to teach - could need a major in elementary ed, early childhood ed., special education , or a specialized content area such as Math, History, English etc. To receive a diploma and teaching certificate the teacher must have completed a supervised student teaching experience. The certificate will clearly identify the areas the teacher is licensed to teach.The state board of education issues these certificates - so they would have the credentialing information. Principals ultimately have the responsibility for placing teachers in the appropriate setting. Could you direct your concerns to the Headmaster? Sometimes teachers do have to fill in temporarily in other areas due to illness or unexpected vacancies. Sometimes teachers can be hired under a "provisional" status. This means they are required to complete the state requirements within a designated time frame. Substitute teachers have no requirements for specific training or experience.
I hope this information has given you a starting point in answering your questions!