In both cases, browsers default to the OS's scroll views - on OS X, that means a scrollbar that disappears, on windows, it means the scroll thumb/bar is always hidden (ever wonder why all scrollable windows look pretty much the same on a given OS?). The only solution, is to NOT use the native scrollable area (overflow:hidden
) and replace it with JavaScript (e.g. http://manos.malihu.gr/jquery-custom-content-scroller/) that re-implements it (but allows you to control the look of) the scrolling interface. Keep in mind that doing this presents it's own set of compatibility issues.
Honestly, the (arguably) best thing to do is just to present users with what they are familiar with - on OS X, that's a scrollbar that gets hidden and on Windows, it's an omnipresent scrollbar. Stick to the defaults and you'll never run into UX issues.