Hi Mary Ellen!
Thanks for reaching out to us! Let me try to help you with this topic.
Unfortunately, there is no built in option to only allow a specific site to import events from your site.
One option would be to disable The Events Calendar REST API from the site you want to prevent the events being exported, and then manually exporting the events as CSV and then importing them in another site using the CSV. But I don’t think this would be ideal.
On the other hand, there might be a way to customize the REST API to require authentication for its requests, but would definitely involve some custom coding, which we cannot give support for.
Is there anything else I can help you with? Let me know.
Best,
Victor