Can a priest and people "dialogue" during the homily?

Answer: PROBABLY NOT
Authoritative: NO

First of all, please note that my answer to this question is not authoritative, which means that it is my own educated opinion and should not be taken as anything more than that.

The question is whether a priest, when giving a homily, may ask the parishioners questions that require a verbal response, or ask them to discuss questions briefly among themselves as thought the homily were a sort of workshop or Bible study?

To the latter question about asking parishioners to "talk amongst yourselves; I'll give you a topic", the only answer I can give is an unqualified no.

The General Instruction of the Roman Missal only says that homilies are encouraged at all Masses and that only a priest or concelebrating priest, deacon or bishop should ever deliver a homily. It also states that a homily ought to be "an explanation of some aspect of the readings from Sacred Scripture or of another text from the Ordinary or the Proper of the Mass of the day " (GIRM 65). These guidelines are found in GIRM 65-66.

The GIRM, however, gives no guidelines on how a homily is to be delivered. I answer "no" to the discussion question simply because such a thing has never been part of Catholic tradition and is foreign to the spirit of the liturgy. This ought to be enough. But, if pressed to offer a "by-the-book" reason why this sort of thing is not permitted, I suppose we could latch on to the definition of content of the homily as "an explanation of some aspect of the readings etc..." It specifically says explanation, not discussion or dialogue, and an explanation clearly indicates one party (the homilist) explaining something to another (the parishioners), not the two parties engaging in a mutual discussion or study session. So I am going to say with a fair amount of certainty that a priest giving his people a topic and asking them to take five minutes to discuss it among themselves is not a licit practice.

But let's circle back to the first part of the inquiry, which was whether the priest, in the course of his homily, can ask questions that demand a verbal response from the congregation. This is a bit trickier, because a priest could conceivably still do this while maintaining the nature of the homily as an explanation.

In the early history of the Church, it was not unheard of for the congregation to offer verbal responses during the homily. Anyone who has ever read the homilies of St. Augustine of Hippo cannot have but noticed the extraordinary manner in which his congregation frequently interrupts his homilies with acclamations, prayers, or lamentations, and which Augustine sometimes responds to them. But then again, Augustine himself never asks for these verbal responses, and Augustine says elsewhere (Book IV of De Doctrina Christiana) that in the late Roman period, it was common for an audience or congregation to demonstrate their approval or disapproval of a speaker by offering verbal cues [1]. Thus the examples of Augustine's congregation offering him verbal responses during his homilies is more to be attributed to the culture of the latter Roman Empire than to any sort of liturgical consideration. In fact, the fact that this was the case in that culture makes it less likely that the practice could or should be adopted to our own. One cannot simply transplant a practice from one tradition to another (see here).

Asking the congregation for verbal responses is a fairly common practice today; even Pope Francis himself has done it. Vaticanista Sandro Magister noted the following at one of the pope's homilies in April, 2013:

"Another typical feature of his preaching is interacting with the crowd, getting it to respond in chorus. He did so for the first time and repeatedly at the “Regina Coeli" of Sunday, April 21, for example when he said: “Thank you very much for the greeting, but you should also greet Jesus. Yell 'Jesus' loud!" And the cry of "Jesus" in fact went up from St. Peter's Square." [2]

Because papal masses are "liturgical paradigms", according to Pope Benedict XVI, it is difficult to say that a practice the pope employs is his own homilies is forbidden in the absolute sense. At best I think we can say this "interactive" homily is a fad, a fad unique to our culture just as spontaneous acclamations were to Augustine's. It is not the how most homilists have done it throughout the ages; it is not encouraged anywhere I can think of, but neither is it prohibited, It is just a fad, one that hopefully will not be of long endurance.

NOTES

[1] For more on St. Augustine's teaching on homiletics, see http://www.unamsanctamcatholicam.com/home/parish-resources/83-parish-resources/321-augustine-wholesome-sweetness.html
[2] http://chiesa.espresso.repubblica.it/articolo/1350508?eng=y