Check with contractor but I think the right side from bottom going up, no handrail on along the landing, after the landing the handrail goes on the longer wall. That’s how builder did mine.

if you are right handed... right side going down.

I do noytthink it matters what handed you are but see what feels natural and sometimes there is only one wall where the handrail can be continuos. I am left handed but only the right side going down could handle the hand rail

Residential code in most areas is both sides unless I a mistake. Unless the stairs are narrower than 44".