If the alarm sounds as someone leaves, call them back through the gate. If the alarm goes off again, then it is highly likely they have something in their possession that is sensitized. If they claim they have nothing and the alarm does not go off again when they come back in and then head out again, it is OK to let them go.
Items that frequently set off the alarm:
library books that have not been checked out;
library books that we have failed to de-sensitize upon checkout;
books from other libraries, including ILLS we failed to de-sensitize;
personal textbooks bought in the Book Shop;
library room or locker keys;
library VHS tapes;
Library Curriculum textbooks.
If the patron has library books and you were not the person who checked them out to them, then you need to check their library account to make sure the books are indeed checked out to them before you simply de-sensitize the books and give them back to the person.