Steve~
Interesting question/topic...
I think (but am not sure) that Long Island (or at least the more eastern Suffolk County) is different - legally - from much of the US. Because settlement was early (1600s) and by grants from English kings - certain rights were expressly granted to original patent holders - and were never superseded by subsequent treaties, etc. Rights to fish (?), shellfish and the bay bottom were - and still are - the province of the Towns - having passed to them from the original grantees. This is why clammers need Shellfish Permits from both the relevant towns (townships, really) AND the State of New York. It is why Pattersquash Gun Club still exists - unique in the US, I believe - with a lease from the Town of Brookhaven. It is also the reason that the South Fork Towns can restrict gunning on their waters to Town residents (Thomas Dongan Patent). Not surprisingly, NYS has contested these patents viz. F&W management - and has lost consistently.
In the subject area (Nicoll Bay - William Nicoll was one of the early patent holders), much of the bottom was leased by the Town of Islip to the Blue Point Oyster Company during my years - where they raised and dredged clams (John Verbeke from the Broadbill film ran their dredge boat for ~ 40 years). As an independent clammer in my youth, I had to stay outside of their staked grounds. I don't know if they held the lease in 1920.
With respect to the Islip Town Gunners Association, I have no specific knowledge - had not heard of it until this past year. I do not know if it had a firm legal basis - or was more of a gentlemen's agreement. Certainly, I have never heard of Islip Town having any involvement with the regulation of waterfowl hunting over my lifetime (which began in 1953).
I will have to check with some others - including my older brother and also a dkbts member who has worked with both USDOJ and NYS DOL.
All the best,
SJS