While you can record the content that is mirrored to Reflector, recording a conversation with these types of applications will exclude your voice. This is due to a limitation of Apple's Airplay protocol that can not record audio from your device and send that audio at the same time.
One possible solution for this type of use would be to ensure that you have enabled the microphone on your computer to capture your own voice, the rest of the conversation will be sent from your phone with Airplay. More details on recording your voice with Reflector can be found here.