Hey Yvonne – unfortunately, I don’t think so. :(
A quick look at Google’s documentation says:
The Private Address lets you view your calendar in other applications like Outlook or Apple Calendar. If you want to let someone else view your calendar, you can share your calendar. You should not give your Private Address to other people. If you accidentally shared your Private Address, click Reset Private URLs to invalidate the old address and create a new one.
So I think if you want to have edit access to the other person’s calendar, you have to use it in “shared” mode.