What good containment looks like
Drop cloths covering the hearth and the route to it, not just a square in front of the appliance. The appliance opening sealed: a fitted cover or sheeting taped in place, depending on the appliance. A vacuum with appropriate filtration running for the duration, drawing air into the sealed space so any disturbed soot goes toward the vacuum instead of into the room. Tools carried in and out through the protected route. It should look intentional before anything is brushed.
Why the vacuum matters more than the sheeting
The sheeting catches what falls; the vacuum stops fine soot becoming airborne in the first place, and fine airborne soot is what ruins a room. A vacuum switched on after the work to clean up has missed the point entirely. The single clearest difference between a sweep who leaves no trace and one who leaves a grey film you find on a windowsill three days later.
The situations that need more care
Light carpet running up to the hearth. Finished built-ins or bookcases beside the fireplace. An appliance in a finished living space, not a basement. Any pellet appliance, because fly ash is finer and more mobile than wood soot. And an insert that has to be drawn out of a fireplace opening across a floor. Each of those is worth mentioning when you book so the containment is planned instead of improvised.
Sweeping from the bottom or the top
Rotary systems working up from the appliance keep everything inside the sealed containment and avoid a roof visit in poor conditions: frequently the cleaner and safer choice in this climate. Top-down brushing has its place. Neither is inherently messier; what determines the outcome is the containment at the appliance, since that is where soot enters the room whichever direction the brush travelled.