Hat collecting can occur across wikis, with users using the rights they've got at one wiki to bootstrap an application on another wiki. This isn't always a problem: most of the user rights require trust. If a long-standing Wikisource admin turns up, there is a good reason to presume trustworthiness compared with a completely new user.
But remember, some wikis are, or were, very lax about handing out user rights. [[Simple English Wikipedia]] used to be known to hand out rollback after two edits... on the basis that the user also had rollback on [[Wikimedia Commons|Commons]]. (That is no longer true.) [[RationalWiki]] is known for giving out admin rights to anyone who isn't a vandal. Likewise, some rights like [[WP:autopatrolled|autopatrolled]] or [[WP:Reviewing pending changes|reviewer]] have many different technical meanings on
[[Category:Wikipedia essays explaining processes]]