(OK, I hate split infinitives:))
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.) Likewise, some rights like [[WP:autopatrolled|autopatrolled]] or [[WP:Reviewing pending changes|reviewer]] have many different technical meanings on the various Wikimedia projects, with vastly different associated levels of trust (some higher than here, some lower). Some even automatically allocate extra rights, like the English Wikibooks, which gives out rollback and reviewer by an automatic process. Be careful not to place too much weight on rights on other projects when considering granting them at English Wikipedia.
[[Category:Wikipedia essays explaining processes]]