Hello,
Typically you TELL the chip WHICH channel you want to monitor, and it takes in samples from that channel only and you get the data out of the data port. When you are ready to read the second channel you tell it to read the second channel, then you can get the data output for that channel from the data port pin. So you either monitor chan 1 or chan 2 and the data you get out is from either chan 1 or 2 depending on which one you told it to monitor. If you want both channels and a little separation in time is ok, you can tell it to read chan 1 and get the data, then tell it to read chan 2 and then get that data so now you have data from both channels.
Typically all the mux does is allow you to select the input channel, either 1 or 2 but not both at the same time usually. If you need both channels read at the exact same time then you usually need two AD converters. I couldnt find your posted part number though ADS128.
Some chips allow you to STORE the data from each channel so you can get it all out later after sampling both channels.