Isolated Ground does not mean how the alternator is hooked up.
Isolated ground is how the alternator is put together. Instead of using the case as a ground, the manufacturer installs a ground post that is "isolated" from the case. They acheive this by using fiber washers between the case and the internal electronics.
For stock wiring, only the passenger side alternator must have this "isolated ground", BECAUSE the wiring system is feeding 12v into the ground post on the passenger side. If the ground post was not "isolated" from the case, sparks will fly because it is just like a direct short across the battery.
The driver side only requires a standard alternator, but they install an "isolated ground" alternator so they only have to stock one type. The reason it has the black ground wire is because the case is not making the ground connection, the wire is.
If the alternators are factory and haven't been screwed up, then yes they can be moved from side to side. This is an easy way of checking the alternator and see if the problem follows it. Sounds like the passenger alt needs a rebuild.
Pleas label all wires before you disconnect them and reinstall in the same location.
The wiring diagrams in the tech manuals show how to hookup the alternators.
Appendix F in the -20 and Appendix E in the -34 (same diagrams).
The alternators have a couple of additional wires hooked up. They are for the ICE/STE diagnostic connection at the bottom of the dash.
