How much weight can screws in drywall hold?
It's always better to attach it to studs - which should certainly be possible with something that big.  That said though, you might be surprised how much a drywall anchor can hold.

How well they work depends on the load though.  They will work best with a static downward load - a fixed weight close to the wall pulling directly downward. 

They will work much worse with loads that pull down and outward (like a shelf) or dynamic loads where the weight changes regularly causing the fastener to loosen up.  Something like a toilet paper holder represents the worst of both worlds - it sticks out from the wall with the weight at the end, and you produce a dynamic load on it whenever you pull on the TP roll.