Other than Virgo, the Sagittarius myth is probably the most commonly misinterpreted of all of the constellation myths.
Most interpretations conclude that the mythology of Sagittarius refers to the centaur Chiron, who was accidentally shot by Hercules (Greek mythology) with a poison arrow. This story does indeed refer to a constellation myth, but it's actually the myth behind the constellation Centaurus, not Sagittarius.
The myth behind Sagittarius actually refers to Crotus, a satyr that lived on Mount Helicon with the Muses. Satyrs have human heads and torsos with goat legs (and sometimes horns).
Crotus, much like Chiron, was a skilled musician and hunter. He even invented the bow, according to Greek mythology.
It's easy to see why the myths behind Centaurus and Sagittarius often get confused. Crotus and Chiron share a lot in common. Both centaurs and satyrs were well known to be wild, rowdy, lustful creatures that had little respect for authority and proper manners. Crotus and Chiron were both exceptions to their races, being instead gifted in the arts and sciences and were knowledgeable and polite to humans. They both were known to hunt with a bow and arrow (though this is a bit misleading as centaurs did not traditionally use a bow and arrow. Satyrs did.) They also look a lot alike. They have the head and torso of a man, but the bottom half of a hoofed beast. Satyrs have two legs while centaurs have four.
Why Chiron gets credit as the figure in Sagittarius mythology is a bit of a mystery. Most likely it comes from different cultural interpretations of the constellations. One could easily interpret the shape as that of a half-man/half-horse instead of a half-man/half-goat. It's just a matter of which stars you think go with which other ones.
Whether you think Sagittarius is based on Chiron or Crotus doesn't really matter, I suppose. Both creatures were very similar to one another and the general meaning of the Sagittarius myth remains the same, regardless of whether you think its representative had two legs or four.