I used 6-sided wood screws, diameter 8 mm, length 60 mm, without dowels.

All information without guarantee.

I used wood screws without a shank for our wooden ceiling.
I chose the largest diameter that would fit through the holes in the roll holders.
I would choose the length so that as much screw as possible holds in the wood without penetrating the other side.
Ideally, the screws should have pan-headed heads. These must not fit through the hole in the roller holder under any circumstances.

Screws are generally screwed directly into wood, i.e. without dowels.
To avoid cracking, I pre-drill the holes slightly smaller than the core diameter of the screw (without thread).