There is nothing to detail. There is no difference whatsoever. "my saying" is the grammatically correct way to say it and is what you will read in books. "me saying" is the colloquial way you will hear it all the time.

are they correct sentences? ex) Do you understand my saying? ex) He understood me saying at the time.

are they correct sentences? ex) Do you understand my saying? ex) He understood me saying at the time.

Please give a sentence example of each. Thanks.