I can't speak for all airplanes, but on the 777 the Air Data Module pitot and static ports had built-in transducers and analog to digital converters that output digital data on dedicated ARINC 429 and on the ARINC 629 Flight Controls Bus to AIMS (Airplane Information Management System) through redundant I/O modules in the AIMS cabinet, and that data was then processed by the displays function for display on the screens, and made available to the FMS and other functions that needed it.
The standby instruments receive direct ARINC 429 inputs from backup pitot and static sensors. The standby instruments are also LCD's, and they can be powered from the battery bus if all other power is lost.
The input air data is processed by the AIMS display system before display. I don't recall the specifics, but I know there was a validation and source selection algorithm, and there are manual source selection switches on the main panel below each ND.