I'm not sure about doing it with distiller.
I used a PDF printer driver and did it that way.  It took me about 5 minutes to write the following macro.  I'm sorry if it gets messed up due to formatting.
Basically, it does exactly what you want.  I had to do the following to make it work.
1.  Set up my FinePrint printer to print automatically to a folder without prompting.
2.  Find out the names of my PDF Printer and my Default Printer (I used the macro recorder for this)
This can easily be turned into a button on the sheet to run the macro, or it could be saved as an add-in.  I would prefer the latter if it were me.
Now, when I run the "PrintReport" routine, it nicely prints each tab to its own pdf with the following names
NameOfWorkbook_1.pdf
NameOfWorkbook_2.pdf
etc.
See if you have an adobe printer driver already installed and if that can be used.  If not, you could even consider a free PDF printer like PDFCreator to use solely for this purpose.  I have PDF Creator at home, and I will see if it works with it tonight.
Also, you can try to just "Record" a macro and do it once.  I'm not sure if it will work the Adobe Distiller though.  (Tools --> Macro --> Record New Macro)
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Sub PrintReport()
Dim ws As Worksheet
'turns off screen updating tabs don't change during print
Application.ScreenUpdating = False
'sets the printer to the PDF printer
Application.ActivePrinter = "FinePrint pdfFactory on FPP1:"
    'cycles through each worksheet and prints each one
    For Each ws In Worksheets
        With Sheets(ws.Name)
            .PrintOut Copies:=1, Collate:=True
        End With
    Next ws
    
'sets the printer back to your default printer
Application.ActivePrinter = "Replace with name of default printer"
'turns screen updating back on
Application.ScreenUpdating = True
End Sub
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