Is there any difference in the requirement for a local law enforcement official signing off  on the purchase of a Full Auto Class 3 weapon as opposed to the purchase of suppressors or short bbl rifles or shotguns? Simply put, do both types of purchases require a local LE's signature before ATF  approval?  

Thanks

The For 4 process is the same for all of them.

Yes.... Or a Trust

You might want to check out the NFA section in the Armory.

But to answer your question.  At the federal level, the BATFE requires a Chief Law Enforcement Officer (CLEO) Signature for all personally owned transferable NFA items (MG, SBR, SBS, AOW, DD, and Silencers) transfered to an Individual.  If you transfer the NFA item to a trust, Coporation, or LLC, CLEO sign off is NOT required.
Now that's just the federal level.

There can be more requirements depending on which state you live in.  Some require the person or corporation to have an FFL, or some require a state permit, like I said it varies from one state to another.

For example:
Louisiana: Localities may have restrictions. State must approve permit
Source: http://www.autoweapons.com/pagelinks/statelist.html

Thank everyone very much for the information.  You've  all been a great help.