Humm,
the active topics island actually should ignore permissions. In fact there is a disclaimer in the cp to that fact.
Even if say a user /guest does not have access to that post the post should appear in the active topics.
Now when they click on it it won't work.
But anyway if active topics does not appear for anyone I suspect it is not enabled.
Either due to you just made the switch and the cache has not had any net new posts in the selected forums to post to the island.
Or you did not complete the required steps to activate the island .
Or you have some permission issues when you installed the product.
Does the island even appear?
What version are you using?
What is the URL to your site?