How do Valentine and Peter resolve the issue of their disagreement about the politics on Earth?

I am going to assume that the question is asking about events that occur during chapter 9 of Ender's Game.  Near the start of the chapter, Peter approaches Valentine about the state of the world.  He sees a great world war coming.  He's afraid that mankind is on the verge of destroying itself, and he wants to stop it:

"Do you understand? I want to save mankind from self-destruction."
She had never seen him speak with such sincerity. With no hint of mockery, no trace of a lie in his voice. He was getting better at this. Or maybe he was actually touching on the truth.
"So a twelve-year-old boy and his kid sister are going to save the world?" 

Of course Valentine is very skeptical.  She believes that her brother is smart enough and power-hungry enough to make a power play of his own; therefore, she is not immediately willing to help her brother out.  
After some more talking, Peter eventually convinces Valentine to help; however, she still doesn't think they can have any impact on the world: 

"Val, if you don't help me, l don't know what I'll become. But if you're there, my partner in everything, you can keep me from becoming––like that. Like the bad ones."
She nodded. You are only pretending to share power with me, she thought, but in fact I have power over you, even though you don't know it. "I will. I'll help you."

The main reason that Valentine is skeptical is because of their relative ages.  They are kids, and she knows that the world won't listen to a pair of kids:

"Peter, you're twelve years old. I'm ten. They have a word for people our age. They call us children and they treat us like mice."
"But we don't think like other children, do we, Val? We don't talk like other children. And above all, we don't write like other children."

Peter's solution to the age problem is anonymity.  The Internet gives them a way to hide their identity and age.  They can write inflammatory and logical pieces that push and pull public opinion in the desired direction.  Peter and Valentine also agree to write from two different perspectives.  Peter will be Locke and write the more rational and empathetic pieces, and Valentine will be Demosthenes and write more hostile and paranoid pieces.  Basically, they are writing the opposite of their own personality.  This forces Peter and Valentine to be inextricably tied to each other.  They can't write their piece without seeking out the advice of the other person:

Demosthenes began to develop as a fairly paranoid anti-Warsaw writer. It bothered her because Peter was the one who knew how to exploit fear in his writing -- she had to keep coming to him for ideas on how to do it. Meanwhile, his Locke followed her moderate, empathic strategies. It made sense, in a way. By having her write Demosthenes, it meant he also had some empathy, just as Locke also could play on others fears. But the main effect was to keep her inextricably tied to Peter. She couldn't go off and use Demosthenes for her own purposes. She wouldn't know how to use him. Still, it worked both ways. He couldn't write Locke without her.