In British English, one can say "our staff do", because they use plural verbal agreement to emphasize when an entity is made up of a group of people, whether this entity itself is marked as plural or not. This is also true of companies, bands, sports teams and other things which are commonly used in plural forms as well as singular forms. The verbs are usually plural for one band or many bands ("Many bands play at the festival" as well as "Radiohead are a band").
In American English, one says "our staff does", because in our grammar, we are not concerning ourselves with whether an entity is made up of many people or not. Since staff is singular, we treat it grammatically as singular. It is no different for us than a stick-staff in terms of grammar.