How long do you have filament sitting "in the open"? I mostly print with a single filament roll at a time, and just leave it on the printer. A 1kg roll lasts me several weeks. For long time storage I keep it in an air tight box with disiccant pouches.
But how long can I let it sit before I should start storing it with disiccant and/or drying the filament before use?
Depends on where you live. I live in a desert where the humidity is often <10% and have left PETG and PLA rolls out for several weeks at a time with no issues. With nylons I've been a bit more careful, though I'm not sure I need to be.
In winter, you won't need to care much as heating will help to keep low humidity, the rest of the year, I would say you can keep pla roll out about a week before storing it but personally (living in 60-85% humidity climate), I just let it out so I'm ready to print. If I start a print after about a week, stringing will likely appear but if stays out for about a month, filament will become brittle.