Help! MacOS & NAS permission errors

We have small office with 3 Macs running El Capitan connected to NAS storage that we self-administer.

NAS is Seagate 4Bay NAS Pro that manages permission by users in groups to dedicated shares.

That works perfectly fine until the point we figured out that some files are not readable or manageable by users in the same group.

Sometimes we get files from our clients or from internet that have something different going on with permissions.

It looks like Mac OS is transferring this permissions from local setup to the NAS. And sometimes Different users in the same group can't re-edit the same file.

Files on the left are once transferred to NAS unreadable by any other user except user who transferred the files to NAS.

Files on the right once transferred to NAS are readable and deletable by others.

first I have tried to get permissions on the left to look the same as one on the right.

So I tried to add this "staff" group in the file locally but I didn't managed to as I can add only "Administrators group"

Please help!