I have a character with two heads but one name. How do I identify to the reader which "person" is being referred to that isn't annoying to the reader?
Are you talking about people with two heads and one body? In such cases, each conjoined twin has his/her own name.
This character is not conjoined, all of her people are like this. One name, two separate heads. When they speak to each other, they just look at the head they are talking too, but that doesn't work when I'm writing because the reader can't see who they are looking at. I think "Left and Right" would sound racists to them.
To 'them'? You were asking how you would identify each head to the reader, not the characters. Left and right couldn't possibly be racist to a reader. They're directions.
I see your point, I thought you wanted me to give them nicknames like "Left and Right" But if I keep using those words over and over again, won't it wear on the reader?
I don't see why it would. I could imagine it being like he/she/they/whatever-else said. It's just a useful piece of info that the reader would not think too much about.
It will get a bit tired if every line points the reader to a head, but once you get dialogue flowing you can use it sparingly and still leave the reader with a clear understanding of which head is talking to which, and you'll mostly just need to address one as the other becomes obvious. 'Looks like rain' the left one said. 'Yep. Damned clouds.' 'Maybe if we grab the umbrella.' 'Umbrella!' the right barked madly, 'that's crazy talk. What are you, a liberal?!'
So it would sound like this? Bob, (left) said, "I prefer my peanut butter crunchy," Bob (right) looked at Bob (left) with a look of distain and replied, "Peanut butter is better smooth, it doesn't get caught in my teeth that way." Bob (left) then rolled his eyes and shook his head in disgust. "I can't believe we have the same mother!"
If they have different personalities (as your example shows), I'd be surprised if they didn't give each other nicknames. I think it's the first thing that an individual learns to differentiate - what's me and what isn't. And a name, a label, is a part of that understanding. Don't they ever talk about each other to other people? Would they say, "It wasn't me who said that, it was him!", without trying to be more specific?
I don't think Douglas Adams bothered differentiating between Zaphod Beeblebrox's two heads, in Hitchhikers Guide to the Galaxy. They were both written "Zaphod", in the original radio script. The differences came out in the writing. Of couse, Zaphod also had three arms.
Nessus, and other Pierson's Puppeteers (from Larry Niven's Known Space stories) had two heads. I didn't matter which one spoke (and he could carry on two separate conversations at once), because they did not have separate identities. The troll was attempting to spark useless debate about distinguish two undistinguished appendages. It makes no more difference than specifying which foot was delivering kicks in an ass-kicking contest. In the case of Zaphod Beeblebrox, the speaker was obvious because the heads were sufficiently distinctive that no name was necessary. And that is the gist of the matter. If it were important to distinguish them, they would have names, or some sort of label to distinguish them, unless it were patently obvious without them. That is how people operate.