And, in the real world, if it were ambiguous, you would likely add "de ella" or "de usted" to clear it up.
I've had conversations like that. lol They sort of go like this: "Ella no me dijo su secreto." "Mi secreto?" "No, su propio secreto. El secreto de ella."
or "Ella no me dijo su secreto." "Ella tiene un secreto?" "Digo no me dijo el secreto de usted ."
Though I've also actually encountered misunderstandings through this very ambiguity when neither person realizes there is an ambiguity and we each take a different meaning. :-/ Usually it is clear by context. But I did have an argument once which ended with someone saying "es suyo" and only later did I realize they could have meant either yours or theirs in the context. Never found out which.