To be more precise, whos department does it land in... and then of course...WHY?
On a large Hollywood style production, the 2nd AC (assistant camera operator) is responsible for bringing the slate. Their other primary duties are loading film (if the production is shooting on actual film), shuttling camera data cards back and forth between the camera and DIT (digital image technician, who downloads the cards onto redundant hard drives), and operating the slate. The slate is generally stored in the camera's front box, which is a little wooden box that goes under the camera lens and is connected to the tripod.
The 2nd AC is part of the camera department. In the camera department hierarchy, they fall below the 1st AC (who pulls focus), the camera operator, and the director of photography (who also oversees the grip and electrical departments).
On a smaller scale set though, sometimes anything goes. On a no or low budget film, it could be anyone.
As for why, I guess I'm not sure. My guess would be that the slate will be used in every single visual shot, whether or not it contains synced sound or is shot MOS (without sound). Therefore, the slate is more important to the camera department than it is to the sound department. Really though, it's the editor who will benefit the most from proper slating later on in post production.
In North America the sound department usually carries the slate as well as any other timecode equipment such as lock-it boxes.
We usually carry the TC equipment because we are the ones that usually jam them from our audio equipment. Of course, most of us soundies believe that the slate should rest upon the camera department since they're the ones carrying the damn thing around. :)
I guess we do it different from the Netherlands. Out here the soundies carry the TC slate, and the Acs carry the traditional slates.
The 2nd AC is also known as the clapper/loader meaning they will handle the slating and camera mags. In Canada, I've never seen a single 2nd AC bring their own timecode slate, and most sound people (even the Holllywood mixers) bring their own slate and sometimes two or more.
I have always seen ACs bring regular slates, and sound department bring the smart slates. Any time a TC lock is required, that is generally generated from sound dpt so all that gear comes from sound.
I bring a slate to any dual system shoot I am on as a sound guy. Both my roommates are camera assistants and they all bring multiples slates each.
It is always good for both the camera and sound departments to be aware of who is equipping what. Rule of thumb - preplanned is prepared, with that being said camera departments are usually the ones bringing the slate - EVEN IN NORTH AMERICA! There are many situations/productions that only use a boom operator (with newer camera's mostly accepting XLR's @ 48Khz) and don't have sound recordists on-hand. Would anyone require a Boom-op to be your slate handler at the same time?
Now if sound is completely separate as in 35mm film and external sound with TC generation - well, special equipment is required due to the nature of that beast - which could fall under either department but most often would be under the sound dept.