The original question included "accurately" and that is the killer for on-board systems in respect of for what the output might be useful.
(a) Typical commercial weighbridges probably will come in at something in the order of 0.5-1.0% and the output might be skewed up to as much as 4-5%.
(b) Aircraft (in hangar) weighing systems are considerably more accurate but, at the end of the day, the end result is constrained by the technology in the particular load transducers and the care with which the weighing is performed. The sort of technology currently available is quite amazing and, for instance, some of the platform kits are "accurate" with no significant calibration error through the usable loading range.
Having weighed something upwards of 2000 aircraft (gliders through to airliners) over the years (with a nominal 0.2% requirement) it became very obvious, very quickly, that, even in tightly controlled circumstances, accuracy (or errors) remains an ever present concern. Indeed, for many aircraft weighing systems, the idea of achieving the normal accuracy and repeatability requirements is, at best, fanciful.
The folk doing the job can only endeavour to constrain the errors to a reasonable minimum. That is to say, when you read your empty weight and CG figures in the cockpit, don't fall into the trap of thinking that they are "accurate" .. they are approximate but should be reasonably accurate.
(c) On board systems should be somewhere between the two, with any sort of luck.
Overall, the on-board systems can be comforting to the crew (I never went to work on the Electra without my on-board weighing kit in my back pocket ..) and, at times, a life saver (we had a couple of major ULD weighbridge weighing errors which were picked up by the system .. and, at least, one which wasn't .. that caused some consternation in the ensuing circuit and landing) but they are unlikely ever to replace in hangar weighing for starting or dispatch figures.
The question of standard weights and related accuracies introduces a whole bunch of other problems but they are not related, per se, to the weighing equipment itself. Certainly the potential is there for the on-board system to pick up significant load build errors, whether they be inadvertant or deliberate.