This is pretty a common misconception. Schengen area only refers to the fact that there are no border checks between two countries which are in that area - it's just like an internal movement. This applies to anybody, so for example if a Canadian enters the Schengen area, they can move to any other country within that area without checks and without requiring an additional visa.

What you're thinking about is the "single market" - which, even if the name could deceive, also applies to people, including the right to go work anywhere within the single market for citizens of single market countries. Which is what many brits always hated, as they wanted single market for services and goods, but not for people.

Single market doesn't imply Schengen though - eg I as an EU citizen have the right to move to Ireland for work, but I still need to go through border checks as Ireland is outside of Schengen (because they have free movement with NI).