If a student practitioner takes notes during background

There are a few things to keep in mind regarding your question, most notably the licensing body of the clinical practice in which you are being supervised. Each discipline as its own national standard of ethics and responsibilities, as well a state laws and regulations. I can speak mainly to psychologists, given that is my licensure. There may be similarities or variations for social work or psychiatry.

In general, all ethical responsibilities and liabilities fall under the practicing/supervising provider. They would be responsible for all documentation associated with the clinical case. I will copy/paste the APA guidelines below for further review, which has multiple guidelines to follow. It is the licensed provider's responsibility to keep proper documentation as it applies to the all patient cases. Then there are personal notes pertaining to a case to be used for completion of the clinical progress notes, assessments, etc. at a later time that do not need to be kept in the patient's records. They are typically disposed of after completion of the clinical notes/assessments.

In regards ***** ***** student keeping personal notes from observing cases, I have not seen anywhere that definitively states whether they can be kept or need to be disposed off under the assumption that no patient identifying information is evident. If there is any patient identifying information, then that is a definite no under HIPAA and state laws to maintain those records. So you are likely fine to keep general notes you wrote about a case, as long as there is nothing that identifies the patient in any way.

If I am understanding the other part of your post, about legally holding oneself as a social worker, psychologist, etc, that is something that each state defines. For instance, in the state in which I am licensed, it is illegal to identify as a psychologist without completion of the state's identified requirements and holding current licensure by the state psychological board. So if I were to stop paying my annual dues for licensure, I would not be able to legally identify myself as a psychologist and to practice in the clinical role of a psychologist as defined by the state. So if you had unidentifiable documentation from past cases you observed while a student, you would not be able to identify yourself as a psychologist associated with that documentation or case.

I hope this answer is helpful. I will copy/paste the APA guidelines for psychology below. Social Work has it's own national guidelines, and I image the AMA has it's own national guidelines whether specific to psychiatry or medical doctors in general.

https://www.apa.org/practice/guidelines/record-keeping