to go after a question together with other people: to talk it over with them or to argue it out against them
The word joins a common verb meaning "to seek" to a prefix meaning "with," and that is the whole picture: two or more people going after the same question. What the word does not say is whether they are on the same side.
Half its uses are people pooling their bewilderment. The crowd at Capernaum turns to one another asking what on earth they have just seen. Peter, James and John come down the mountain trying to work out what rising from the dead could mean. The two walking to Emmaus are still turning over everything that has happened when a stranger falls in beside them. In each case nobody in the group has the answer yet, and the talking is a search.
The other half is a fight. The Pharisees come out to Jesus and start pressing him for a sign from heaven. Religious teachers stand in the middle of a crowd arguing with the disciples. A synagogue crowd in Jerusalem takes on Stephen, and the newly converted Saul takes on the Greek speaking Jews, and both of those arguments end with someone trying to kill the speaker. The word is the same; only the temperature has changed.
That is what sets it apart from διαλέγομαι, which pictures someone laying out a case and working it through with an audience, and from διαλογίζομαι, which can happen silently inside one person's head. This word always needs another party in the room, and its accent falls on the open question being chased rather than on the case being built.