Did Judaism really draw beliefs from Zoroastrianism, or the other way around?

I think in some ways maybe Judaism did borrow from Zoroastrianism, and in other ways not so much. Remember, Zoroastrianism is dualistic, Judaism isn't.

Zoroastrianism was actually the first monotheistic religion, eventually Zoroaster suggested that every God except for Azdu--Mazda were demons pretending to be Gods.

Zoroaster came first.

Zoroastrianism and Judaism proper developed at about the same time. The Avesta shares some of the same prohibitions as the Torah. One that I remember is the prohibition against a man having sexual relations with a menstruating woman. It's impossible to tell whether one borrowed from the other.

Zoroastrianism is older, so I'd guess that they had the original ideas.

Judaism was profoundly affected by exposure to the Persian Religion. Prior to the Babylonian exile Judaism expected everyone to go to Sheol following death. Sheol was a place of peace, with all the pressures of the outside world gone. There was no heaven or hell, only Sheol for good and bad alike.
Judaism also taught that God ruled the world through a Divine Council composed of Angels. Each man came before the Council annually to find out if his name was included in the Book of Life. "The Satan" was a member of that council who God depended on for reports on any wrong doing. "The Satan" roamed the world in the performance of this job.
After exposure to Zoroastrianism Judaism embraced an afterlife, resurrection, and yes, even dualism proved by "The Satan" becoming Satan in the later writings. Satan is no longer a trusted member of the Divine Council, but an adversary of God. Evidence of the change in Judaism's attitude toward Satan comes from comparing a passage from the Book of Kings that God caused King David to order a census among his subjects with the version from the Book of Chronicles: Satan caused David to order the census.
Chronicles is generally accepted to have been written by Ezra the Scribe who also receives credit for the Book bearing his name and possibly Nehemiah as well. Ezra was a Babylonian Jew who was active around 300 BC. He is also believed to be the final editor of the Old Testament, responsible for its present form. Ezra the scribe is just as important to the Old Testament as the Apostle Paul is to the New Testament.

Jewish beliefs came from G-d.