There are a number of good answers about writing the scene on here, but I feel that the meaning in the original question are getting lost. I believe the question is how to avoid so many "He says" "She said" "He said back" fillers.
In my opinion it is good to mix in other descriptive words such as "He says angrily." Or use phrases such as "She replied with a smile." I believe it accomplishes the same task without the noticeable repetition of "He said" Over and over"
As an example here is an excerpt from my work:
“Where’s my mother?” The newcomer demands and reaches inside his long coat, his hand emerging holding a pistol of his own.
“I know not what you are speaking of.” Kuto answers, fully drawing his own gun and pointing it at the stranger.
“Don’t be a fool! Grampa told me someone took them from the camps. You will surrender them to me, brother, one way or another.”
“You have mistaken me for another. “ He lies—he knows this man is after the two women and is likely trying to confuse him as if they were related. Now the two men are facing each other, pistols pointed at the other’s heads, circling slowly. They stay this way half a minute, neither talking, and neither shooting, until a third voice breaks the silence.
“GABRIEL!” Monica shouts as she runs out of house she had dove into when the chopper was still on approach. The young man turns and drops his gun to the dust. His face immediately softens and he runs to meet her as well. They embrace deep and long, parting only after neither can breathe.
“I was so worried Ma. Gramps told me what happened. Are you okay?” He stutters, worry clearly still clinging to his voice.
“I’ll tell you more later, but all you need to know now is I’m safe… well me and Natasha… thanks to this man here.” She says, gesturing to Kuto with her hand. He turns and looks the native in the eyes.
“Thank you then, brother. I owe you more than I can repay.” He says.
“Aye. My family is killed by the same man that captured your mother and would not have been involved if not for their relations to your family. But you are welcome for what can be excused and for helping your mother.” Realization hit Gabriel when he looks past Kuto and sees the three fresh graves as well as the smoke still rising from the home.
“If it helps, I am now making it my personal mission to find this man that hurt my mother and bring him down.” Gabriel says.
“Tis my mission now too. To find and kill this thing—for he is no man—to avenge my people.” Kuto says. His normally bright honey eyes now harden with determination.
So you see. I do have some "He says" Over and over but they are broken up by some other descriptive phrases and actions accompanying the statement or reply.