The demands made upon the director, as well as any of the dept. heads in pre-, production and post, will vary depending upon the budget level.
I do lots of work at the low/no/mini/micro budget level. I do all of my own cue sheets for dialog, Foley, sound effects, and score/music. I do all the dialog editing & clean-up, perform all of the Foley, create from scratch and pull from various sources all the sound effects, edit the score & source music as needed and mix the results. I very rarely get involved until the final cut (or close to it) is complete.
At the "Hollywood" budget level the supervising sound editor oversees the teams for each of those categories (DX edit & ADR, Foley, sound FX) so that everything comes together as a cohesive whole during the mix. Sometimes the sound designer is also the supervising sound editor, but not always. The sound designer and/or supervising sound editor may or may not be deeply involved in preproduction. Sometimes the sound designer and/or the supervising sound editor will mix, sometimes not.
And there are many shades of gray in-between.
Production sound is the same. At the indie level it's one person with a basic kit; at the mega-budget level it's a production sound mixer, a boom-op and an audio assistant with all the variations in-between.