In ficion today, we add she said and similar dialogue tags only when it is unclear who is speaking. Otherwise we avoid them.
Turntaking (that is, when the speaker changes) is signalled by a paragraph break, e.g.:
"What are you talking about?" Peter asked.
"Nothing," John replied.
"Then shut up." ← Peter again
"Okay." ← John
All the dialogue in one paragraph is by one speaker:
"What are you talking about?" Peter asked. "I don't understand you." He turned away from the window and walked over to his brother, who sat with his head in his hands. Peter shook him violently. "Answer me!" ← all of this is spoken by Peter
John remained silent.
"You disgust me." ← Peter again
After several turns, you may need to remind your readers who is talking. Also, you may want to add descriptions of what people do while they are talking or how people speak, e.g.:
"What are you talking about?" Peter slowly turned away from the window.
"Nothing," John replied sullenly.
This will make your dialogue more interesting and less monotonous as pure dialogue (without any speech tags) or repetitions of he said / she said.