I use two computers for giving lessons. Some of my students use two devices, typically a phone for the video and audio, and a computer or tablet for reading articles. I make (or receive) the call from my preferred computer (with webcam). On the second device, I select the same user for Skype chat. Now I can send text or documents to the person from either comptuer. Usually, I share the screen on the first computer and display an article to discuss or a corrected text. I can highlight sections of interest. On the second computer, I make discussion notes in the chatbox, and at an appropriate time, I pause the lesson to send the notes and then discuss them.

My advice would be to use a GoogleDoc to write down corrections or new words. I only use the Skype chat in trial lessons. All my regular students get a GoogleDoc. We can both write on it, it's instant synchronisation, students can use it for their homework and add comments or questions.