Is there a word or phrase that describes a statement that implicitly requires the audience (a person or persons) to respond with a question? For example, if someone says to you, "I just saw the best movie ever!" You, in turn, are implicitly required to ask, "What movie did you see?" The idea I am looking for would be the antonym of a "rhetorical statement", if such a thing existed.
I believe this is simply the nature of conversation. Asking a question is only one of the acceptable ways to move forward in such a scenario. The audience may respond with a guess instead of a question. The audience may pause and wait for a follow-on statement which provides more information. The audience may also choose to ignore the statement entirely and change the subject.
To my knowledge, no such phrase is commonly used in English. The sole exception would be perhaps a leading statement or question. That is not to say that it comes before something but rather it prompts the user to subconsciously respond (though not necessarily with a question).
Many attorneys (or at least the good ones) are proficient in this as they can trick the respondent into revealing more information than intended.
There could be such a phrase used by academicians in the field of linguistics however.
There is a slang term for Facebook status updates that are like this:
An intentionally vague Facebook status update, that prompts friends to ask what's going on, or is possibly a cry for help.
See also: Vaguebook.
In real life, I would call it a vague, attention-seeking statement. (But not to their face.)