How can I show next/previous paging that's limited by specific criteria?
You can only pass an object or an ElementCriteriaModel (what you get when you type 'craft.entries') into getNext() and getPrev(). 

Calling find() on an ElementCriteriaModel, as I'm doing in my code, turns it into an actual array of entries, which is not something getNext() and getPrev() can work with. 

So changing the first line to this will work:

{% set params = craft.entries.section('leadership').leadershipType(entry.leadershipType) %}


Or this using the object syntax for setting parameters:

{% set params = craft.entries({
    section: 'leadership',
    leadershipType: entry.leadershipType
}) %}