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I'm getting conflicting information as to whether the two halves of a P-bass pickup are configured so that they are noise cancelling, so
Are the pickups in standard P-basses (say a Fender American Standard if we need a specific model) designed to reduce/eliminate external noise?
Yes. The "split coil" pickup (actually two separate pickups placed close together under the strings) is a humbucking, noise-cancelling design. These have been standard on the Fender Precision Bass since the early 1960s.
Split-coil noise-canceling design
However, the original "vintage" Fender Precision Bass pickup is a single pickup with a single coil, and it is not noise-cancelling -- it is subject to 60-cycle hum electrical interference.
Original 1950s P-Bass single-coil pickup (not noise-cancelling)
The original single-coil design is less popular, but it is favored by some traditionalists, notably Sting. Occasionally Fender will re-issue basses with this "vintage"-design pickup.
Will's answer is precise, however keep in mind that this transformation of the Precision Bass really has lead to a few types of pickups common for P-Bass installation.
Different off-the-shelf P-Bass models could use traditional, 60-year old single coils, a split single coil that may or may not be hum-cancelling through reverse-would reverse-polarity wiring, straight beefy humbuckers (sometimes using two Jazz Bass pickups stacked together, also reverse-wound from each other) or even a jazz bass pickup that may be noiseless or not.
