Hi Nic,
It is indeed possible to display events in separate calendars on your site. Your idea of using categories is a good one – to get two separate calendars set up quickly and easily, you can use the tribe_events shortcode with the category filter to display each calendar wherever you would like on your site (instead of using the main events page).
To exclude a category from the main events page, you would need to dive into the code a bit…this post has an example snippet that you can tweak to exclude your categories for the views that you need to exclude them from.
Additionally, this article has some other tips on making a “members-only” calendar that you might find helpful.
Please let me know if you have any other questions!
Thanks,
Jennifer