Was in an accident. I thought it was mutual fault and that we would just pay for our own cars damage. They are asking for cash from me now. What should I do?

The accident took place in Washington State. It was in an Intersection with 2 left turn lanes ( like this http://i.imgur.com/H2at2Kb.jpg ). The intersection was just paved and there were no turn lane guide lines. We both just gradually came together as we turned. I hit his back left bumper with my front right bumper. We exchanged phone numbers and licenses but not insurance information. The other person was in a big hurry and said they would call me about it later. They said it was a small dent and wouldn't cost much to fix. 1 week later they said it would cost them $1200 to fix and that they could manage if I just gave them $500-$600 cash. I told them I would think about it and get advice on what to do. Another week goes buy and they text me saying the damage is 1500+ now and are asking for $700 in cash by next week. I've never been in an accident so I'm sure I made some stupid mistakes in this exchange. Going forward what should I do?