If you are just doing a couple, then this should not be a problem.
Whilst the majority of our herd is outside, we farrow in sheds so are familiar with your circumstances.
Presuming your sty will have an inside and outside, ensure the inside is big enough for a bed, but not too much bigger than this. In essence they will then create a nest area where they sleep - everything else is them available for eating and pooing, and if the inside is large, they will get out of bed walk one yard and then just pee/poo. When you get your weaners you may want to make the inside smaller (say with an uncut straw bale or two) to stop them doing this. So create a small inside and larger outside, and most of your probelms will disappear. If inside area is small enough that they are only using it for sleeping, this bedding will not need cleaning very often - maybe fortnightly in winter and once a month in summer, with suitable top ups in between. With the poo area outside if this is concrete you should "poo scoop" once a day, pigs do not dung on top of dung, so there poo area will just increase if you don't.
If it is outside on earth, you may not need to scoop at all, or only during dry spells - just watch and see if it is being washed away by rain and dragged down by the worms. This should go on the compost heap. 2 weaners through to slaughter weight will not generate masses, and a small compost heap will be all that's needed.
If you are entirely indoors, then they need three areas, a next area, a pee/poo area and a feed area - they will decide which is which. Pigs are quite clean, and you should not need to change their nest straw very often, so you will not generate lots. What is generated can go on the compost heap. With the poo area inside a daily scoop is essential, and if done regularly will keep the area very clean. Pig poo should be firm (if not you have a health problem) so scoops very easily.
Which just leaves peeing and washing water, which is the essence of your question. If entirely inside, the pee will generally dry, and if you poo scoop daily, you will only need to wash down to clean the smell of wee. This is entirely subjective, but would not expect to wash more frequently than weekly in most circumstances, and to need no more than one bucket of water (no detergent, plain water will do) - this can just run off to whereever, it will not be strong enough to do harm.
If pigs are peeing/pooing outside, again scoop daily, and washdown only if on concrete as per inside. If on earth would not expect to need to washdown.
You will generate so little "liquid" effluent with 2 pigs that you don't need to worry.