It depends on what you mean by "just notifying the bank," but if they have not provided any type of request for information about the mortgage loan, then there is no obligation to follow the request process, as nothing is being requested. However, if you are a "Large" servicer, then you need to have policies and procedures that deal with prompt facilitation of communication with the potential successor in interest. You would also need to promptly determine the documents needed to confirm the Successor in Interest status and provide a description of such documents to the potential SII, along with an explanation of how (and to what address) they can submit a written information request under 1024.36(i).
Although, a small servicer seems to be able to technically do "nothing" in the exact scenario you mentioned, as there is no request for information, it may be a good practice here to provide a description of the documents required to confirm the SII status and contact information for further assistance.