For standard ABS and PLA filament, most distributors recommend storing the filament in an airtight bag. Does not doing this actually make print quality worse? I have left mine in the open for a year and have had no noticeable problems.
It makes a difference where I live, and I'm not in a particularly humid climate (California). When printing with wet filament, you'll sometimes hear it popping and see steam coming out of the extruder (it's usually only this extreme with nylon). With most other filaments, when they're wet, the extruded filament will have small bubbles in it and the surface finish of the parts will be rougher, with breaks in the layer lines. It can also lead to more oozing and stringing. Air print a few centimeters of filament and look at it closely to see if there's any bubbles, if not, it's probably dry enough. Whether the filament absorbs enough water to be noticeable in a few hours, in a day, or in a week depends a lot on the filament (and I assume the humidity too). I'm mostly noticed problems with nylon, ABS, and NinjaFlex, less with PLA and PETG (though I avoid leaving any filament out for more than a day).
If you're not seeing any difference between, then I wouldn't worry about it. Storing filament dry is a hassle.
Humidity may be the problem.
Humidity tends to degrade filament, making it weaker. If you leave a coil of filament out, over time it will be exposed to humidity. I have yet to hear of this happening over a short period of time - the real threat comes if you leave it out for weeks or months - but it can happen nonetheless.
Contamination with other materials is possible but unlikely. The odds of some sort of impurity developing from nearby particles is extremely low unless the filament is actively exposed to some other material.
In most cases, though, things should be just fine.
In most cases, you should be fine with ABS or PLA out of an airtight container. If you're worried about it, throw a few desiccant packets where you store your filament.
However, some specialty filaments should be stored in an airtight container. PVA is notorious for absorbing the ambient humidity around it. When it's heated, the water it has absorbed starts to form bubbles, completely messing up the extrusion.
In short, some specialty filaments, definitely. With others, it isn't necessary, but it can't hurt.
I haven't had any other issues storing it in the open, but keeping it in an airtight environment (especially if you live in a humid environment) keeps it moisture free, which can effect print quality. Manufacturers recommend this to help keep filament dry.
Normally you should check and follow guidelines provided by filament manufacturer or please contact them for more details. It can vary as not all environments are the same. For example frequent temperature fluctuations can increase chances of making a filament turn brittle.
PVA specifically is a water-soluble filament, so keeping it dry is absolutely necessary, because when it gets wet it can become unusable. Therefore when possible, keep them in original packaging (a sealed bag with a dessicant packet) as long as possible.
See: How should I store my filament? at MatterHackers FAQ