On a lathe it tends to fall through or on the opposite side of the carriage moving direction so it is not an issue. As to grinding it is don at low speeds and most cover their bed with a cloth to collect it. Changing chucks I was taught lay a small section of plywood on the bed to prevent a ding if the chuck slips from your fingers during the change. Finally most drive components are out front, protected, easily inspected and cleaned.
Mills on the other hand generally have motion screws that are semi protected and not accessible for cleaning and inspect. Ways tend to collect scarf and and during motion (mills in comparison move a lot more) tend to jamb scarf underneath the slides, up until recently I did not use covers on my mill because it was easier to clean constantly when it was a manual mill, now that its CNC I use bellows type covers (McMaster-Carr) between at the column and slide side, formed metal covers everwhere else. I still need to inspect and clean underneath regularly (daily during use), with the bellows cover having to be cleaned every 1 or 2 parts depending on the operation due to scarf build up.
3 months of machining resulted in about 28lbs of Aluminium scarf, yes I machine a lot.
Even true modern CNC machines need help in keeping scarf under control with human intervention.
