One of the downsides of a weekly-typed language is that you have to jump through lots of hoops if you want to protect yourself from every concievable ill that clients of your APIs can heap upon you.

So, how much testing should one do to see if arguments are what are expected, and how does "overloading" functions to understand different data types complicate matters?

Do as much testing as you need to, though if you find yourself using isa a lot, that's an indication that your interfaces need further thinking. A technique from Smalltalk, which I've seen used a bit in Java, is "double dispatch". It's a technique that, in part, lets you contain a geometric explosion in type testing when you introduce new types of objects. Look it up and study it. Double dispatch can be done in Perl.