For safety reasons, a magical museum might well want to avoid putting actual items on display, especially since Illusion magic makes it trivial to create simulated substitutes crowds can safely interact with. This also keeps security budgets manageable, since even a magical museum probably doesn't have the kind of money to protect the arcane equivalent of a small nuclear arsenal.
So the museum researchers might be happy keeping items on indefinite loan, as long as they know the borrower and have guaranteed access when they need it. This way the mage owning the item is responsible for keeping it safe, and arcane historians--who definitely didn't spend all those years in magic academy just to play guard--don't have to.
As a bonus, this method gives the party a quirky but knowledgeable NPC contact who can give them clues or set up sidequests whenever you need.