My Editor is Pretty Strict:
Every time the discussion is switched back and forth, it either needs to be understandable even to someone who hasn't been paying attention as to who is talking, or else you need to say who is speaking. So very short clips of speech can be stand-alone, and then only if there is NO possible way someone else said it.
Conversations between two people of the same gender make he/she problematic, so don't rely on it. Even then, he said/she said is terrible form. He can quip, she can mutter, he sighs and she yells, but you're right, saying said is dreadfully boring. Colorful language goes a long way towards breaking up the monotony.
Also, even if you can use a he/she, don't do it for two times in a row. So don't say he did, then he said, then he did, then he said. You'd say Ted did (making it clear who is involved/speaking and there isn't another guy in the conversation), then he spoke, then Ted did (again, reinforcing it is Ted) and finally he spit the words (being colorful).
But you can inform who is speaking in ways other than telling. You can describe what a character is doing or thinking, and in the same paragraph infer that that it is them speaking. Don't use dialog where one person speaks and someone else makes an action in the same paragraph. You can say:
Ted wrung his hands. "I'm really sorry. How was I supposed to know?"
Jan couldn't look at him. "I was drunk. I couldn't even stand up. What made you think it was okay?"
This way, you give a visual, or an insight into the character's thoughts (showing without telling) while still making it clear who is speaking. If you said in one paragraph:
Jan couldn't look at him. Ted said "I'm really sorry."
It muddles what Jan and Ted are doing and saying.