Assuming I start ArcGIS 8 times on my PC. One for each processor on my PC. Now lets say I run similar but slightly different models in each ArcGIS. Lets say reclass, zonal stats, or math. Now assume all 8 have the same inputs and outputs but the process is different. Could I get cross-pollinated results with all 8 using the same scratch directories or are the temp files tagged with the processor ID or similar to keep them distinct?
With instances of ArcMap and ArcCatalog no, each one creates a temp session directory in your %TMP% folder called arc<4 hex digits> like arcA7B0 and is thus unique.
For instances of python, yes, it happens frequently. Multiple running instances of python scripts started from command line will cause each other to crash randomly. This limitation does not apply to python scripts being run from ArcCatalog/ArcMap but only to scripts executed from the command line (cmd) or by batch file.
To verify this open the python window in ArcCatalog/ArcMap and type:
import os print os.environ.get("Temp")
To verify what your temp directory is open a command prompt and type:
Also, you can see them in Windows control panel / advanced system settings / advanced tab / Environment Variables button... but it's easier to echo (print) them.
Set T also will show any environment variable starting with a T in DOS.