AFAIK, to date Twitter hasn't required its users agree to a copyright assignment or an exclusive license to the content of their tweets, so it wouldn't be a cop

AFAIK, to date Twitter hasn't required its users agree to a copyright assignment or an exclusive license to the content of their tweets, so it wouldn't be a copyright suit. Twitter wouldn't have standing to bring suit, as that would be reserved for the actual copyright holders, i.e. Twitter's userbase.