The iOS Human Interface Guidelines say:
Use the system-provided Share button. Users are familiar with the meaning and behavior of this button, so it’s a good idea to use it when possible. The main exception to this is if your app does not contain a toolbar or navigation bar[, as] the Share button can only be used in a toolbar or navigation bar.
OK, but how do I “use the system-provided Share button”? A search of the documentation turns up nothing useful.
I've gathered that I should use UIActivityViewController in my response to the button being tapped, but how would I create the standard Share button in the first place?
The standard Share button is a UIBarButtonItem (so it can only go on a navigation bar or toolbar). You need to create a “system item”; specifically, an “action item”. The “action” bar button item is the Share button.