Is it necessary to inform the contractor when a worker undertakes private work for the engineer who was responsible for the building, or does the worker have the right to choose?

There is no need to seek permission from the contractor to accept work offered to you after you have finished your work with him. The contractor's right is limited to the duration of his hiring of you. There is no blame upon you for accepting other work from any party, even if you became acquainted with that party while working with the contractor.