Relevanssi is an excellent plugin that replaces the standard WordPress search with a better search engine.
PublishPress Permissions Pro has integration with Relevanssi. If you use PublishPress Permissions Pro, your Relevanssi search results will have the correct visibility.
What does this mean in practice? Imagine you set a post to be visible only by people in the Administrator and Editor role. If you have PublishPress Permissions Pro, then Relevanssi will only show that post in search results if the viewer is an Administrator or Editor.
To make this integration work, you don't need to do anything after installing PublishPress Permissions Pro and Relevanssi. However, if it's not working, go to Permissions > Settings and make sure that the “Collaborative Publishing” box is checked: