The other person can’t see me on their webcam, even when I have mine turned on. I can see them though.
Tell us about the details of your OS and vendor of Webcam with model.
If you can see them and yourself in the small window then they need to configure Skype to work with their webcam. Ask them to post exact details of what they are or are not doing here and we will try to help them out.
Can you see yourself in the small video window? Check out these troubleshooting pages on Skype help: