Got it to hide using CSS:
.CommunityPage .search-granularity { display: none;}
Better way - specify the element within the body of the community page:
#lia-body.CommunityPage .search-granularity { display: none;}
Now that we know the "how," I would be interested in the why. In particular, removing the user search means that people have to click the search button with no entry, or an irrelevant entry to get to a page where they can specify that they want a user search. That is confusing and many won't figure out that process. Additionally, if you add our Tribal Knowledge Base feature at some time, you can also focus your search there.
The answer is related more towards the feature of this search functionality. We are using this component on our landing page and customizing the design to fit our style. What we wanted was the initial search box with type ahead functionality and have the user execute a general search. We found that it wasn't necessary to have the users selected a category from the dropdown, so we wanted to remove only from the home page (which is why you see the CSS element to hide it only on that page).
To add to my previous request, I would add that you should not only be able to remove the search granularity but also specify which filters to apply (i.e. search in Ideas or Blogs).
