A great discussion, and all true. Although the Polish may be entirely correct and natural, the resultant English is not. Perhaps this might be one of those times where one cannot use a literal word for word translation, but rather must translate the thought and intent for this to be correct in both the source and target language.
AnCatDubh meant "Auntie" works because it can function as a nickname. We can say "Have you visited Auntie?" (without determiner) because Auntie functions as a proper name, but we can't say in English "Have you visited aunt?” because it is lacking the obligatory determiner (a, the, your, my aunt, etc.), and because "Aunt" would be very, very strange to use to refer to someone except when directly speaking to them.
The lack of determiners in Polish where English speakers expect them is kind of hard to get used to, but the occasional dropping of pronoun determiners (twoje, jej, etc.) is even harder to grasp because they are sometimes required. It's hard to tell when it can be dropped or assumed, when you start out linguistically programmed to never leave them out! That's all. :)