Perhaps some one more up to date than I am can step in and correct me if I am wrong. I am not sure about HR218 in a case like this, it seems like it could apply. If it applies to this position HR218 would supersede NYS law as I understand it, but in this case I am really not sure how it works. I would strongly suggest any member of that dept. consult with this agencies firearms instructors before carrying in the city. Seems like this could be a grey area.
In NY law these positions are probably not considered exempt. In the NYS CPL- 2.10 for peace officers if you are "exempt" that means you can carry on a badge and do not have to carry on a permit. So my understanding is if under the NYS def. for peace officer there is a line saying "that nothing in this subdivision shall be deemed to authorize such officer or agent to carry, possess, repair, or dispose of a firearm unless the appropriate license therefor has been issued pursuant to section 400.00 of the penal law; and provided further that such officer or agent shall exercise the powers of a peace officer only when he is acting pursuant to his special duties."
If you see this after the title used in Westchester Co. then its my understanding you would not be able to carry in NYC, because your title lacks authority to carry on your badge. They can allow you to carry a firearm while on duty, provided DCJS approved firearms training is completed, but you still would not have statutory authority to do so without a permit. So in theory this would also mean no AW or hi cap mags.
NYS has some strange laws and exemptions. In the CNY area you have Onondaga County Aux. Police (peace officers, no firearms), and Onondaga County Special Deputies (not peace officers, carry firearms). Go figure.