This question can also apply to bands, but for the sake of the discussion let’s limit it to being about solo artists.
In most professional artists’ polished studio recordings, there are many instruments that you might not even notice until after a number of intent listenings. When the artist (singer-songwriter in this example) writes the song, do they just write a melody/chord progression and someone else takes care of all the rest, or do they arrange all those other instruments too?
It depends.
I suppose it would depend on the artist. Some songwriters/performers want to decide every little detail of a song, laying out all the instrumental tracks and arranging everything. Others (I’m sure I’ll be one of these others, once I eventually get that big recording contract and fancy-schmancy producer to work on my album… 
) are content to come up with a good basic song, melody, lyrics, chords, etc., and let the producers and studio musicians flesh it out. So it really would vary depending upon the ambition and control-freak-factor of the artist.
Also on the talent of the artist. I have worked on CDs for two local artists.
One is tremendously talented, with a world-class voice and an ear that is super-human. She writes every note of every song–bass & drums, piano, orchestra–and then calls me in to play guitar solos and add some background vocals. I try to play some creative stuff and sing in tune.
The other writes folk-ish singer/songwriter stuff. The tunes are fairly predictable but pretty nice. She can barely sing–and not particularly in tune. I ended up being the producer on that CD, doing the arrangements, finding the players (bass & drums), etc.
It varies with bands, too, although typically the bass player creates the bass part, drummer the percussion and so on. There are exceptions: on one Cars album (remember them? the boys from Boston) the producer and the keyboard player sequenced all the drum parts (that was innovative at the time) and the rest of the band played around the recorded tracks. Then when it was time to tour, they called in the drummer to LEARN the tracks. He was not amused.
I guess the question is: can you afford someone who will be more creative/a better player than you are?
I think, and this is my opinion, but someone like Sting - who was a Music Teacher (and can play several instruments), before the Police (maybe the Police was more collaborative) took off, but I think Sting writes the entire song, Lyrics, The Music (Bass Line - off course), Lead Guitar, Keyboard, Drums, other instruments. This is why it takes him longer to record and get new music out, but the wait is worth it. He has re-done/re-worked songs in different ways for different concerts - videos, etc. I think this is why he left The Police and has never went back. He likes the control of creating and producing what he wants.