What's the difference between 両方 and 双方? I am definitely more acquainted with 両方, so I guess it is used for a wider variety of things, but could anyone tell me the specifics of what they can or cannot pertain to?
I think your guess is right that
両方 has a wider meaning.
双方 means "mutual sides," so it is used only when the two objects are in some relation to one another. You can use either one when there is some relation between the two objects (people):
'hear about the case from both the victim and the perpetrator'
'hear about the case from both the victim and the perpetrator sides'
but when there is no particular relation between the two objects, it is awkward to use
'borrow class notes from both Tanaka and Suzuki'
'borrow class notes from both Tanaka and Suzuki sides'