"if a method needs to handle an exception" - are we talking about checked exceptions here? Because you never actually need to handle an unchecked exception; to put it another way, it's generally best to prevent unchecked exceptions by making sure you don't pass illegal arguments into functions.
If a function declares a checked exception, the function is probably part of a library, and the exception is probably intended to be handled locally by the method calling the library. So yes,
you should handle it locally, unless for some reason the design of your application is such that it's intended that the exception be handled in a higher layer of code.