Not seeing yourself or the other person at the start of a call is a common problem among all video software. Here are a few things you can try:
Restart your computer<-- THIS SOLVES MOST ISSUES Sometimes other video programs (e.g. Skype) can adjust your settings. Restarting your computer will clear this issue.
Did you give your browser permission to access your webcam?
The first time you log into Doxy.me, your browser will ask you permission to share your webcam and microphone. Make sure you selected the correct webcam to share with Doxy.me.
Run this test to check your webcam capabilities.