Even though the Republic is technically a "dialogue," we hear all about this dialogue from Plato himself. Now, Plato must have had a killer memory to keep track of this 300-page discussion—assuming this is even an accurate transcription of a real dialogue that actually happened, which is unlikely.
While Socrates doesn't spend a lot of time sharing his feelings, we do get a sneak peak at his anxiety during his argument with an angry Thrasymachus: "I was astounded when I heard [Thrasymachus], and, looking at him, I was frightened" (336d).
Socrates's (and Plato's) potential unreliability as a narrator also comes out a few times when he reveals that he's been making some edits in how things happened: "Now, Thrasymachus did not agree to all of this so easily as I tell it now..." (350d). It's possible that Plato is reminding us that we're reading Socrates's version of events and is inviting us to not necessarily trust every detail he describes.