Getting a residence/work visa for non-EU citizens (Stockholm)

Hello all,

I am new to this forum. I currently live in another EU country and have a residence visa for that country. I hope to get Permanent residency in a few months and once I do, I'd like to relocate to Sweden. According to what I've read about residence rules, once 1 gets permanent residency in an EU Schengen country, 1 is allowed to work in another EU Schengen country. Is this correct? What is the job situation for language teachers in Stockholm?

Thank you.