No group permissions?

That's somewhat normal and should be expected.

It is likely that Word is saving the file without a group as other programs (like TextEdit) will add a group entry. Word 2016 did not when I tested.

Why does this cause a problem with DropBox? It shouldn't care what group the file belongs to.

I would check the value of the umask environment variable.

To check the value, open Terminal application and at the shell prompt type umask.

(That is, type only the word umask and hit enter.)

I suspect that your file has no permissions set for group and therefore non are printed by 'get info'.

BTW - my umask value is set to 22.

read about setting umask values by typing man umask at the Terminal shell prompt.

Here's information about setting umask on OS X.

Setting a custom umask in OS X - Apple Support

Since you are not using a true file server to serve files, the best way to handle this is to create ACLs on the folders which will allow all members of the group to modify the files you store in that location.

See this thread: Re: Granting full control for every user on folder Users/Shared?

You don't need to modify stuff on a per-user basis. You just have to add the users to the group.