I'm guessing people will live in cheap places that are considered to be in the same region as expensive places. For example: live in Tracy or Fairfield or Hollister while collecting a bay area salary. Or live in Vancouver, WA while collecting a Portland salary. The latter is especially enticing as Washington has no state income tax.
In the long term though, if companies go remote I don't see how they can continue to have salaries based on cost of living. In a hypothetical perfectly frictionless economy with perfect competition, companies that pay salaries based on the value contributed by the employee would be more efficient and outcompete those that paid based on the employee's expenses.
The number of engineers isn't changing. If a bunch of engineers leave silicon valley, and some companies decide having people onsite is still critical, the bidding wars for the engineers who remain goes higher. All in all, I don't think much will change. If engineers refuse lower salaries, they simply won't leave the bay area -- keeping things in aggregate as they currently are.
How can/will the employer check whether the stated location of its emloyees is correct?
This is indeed a complex categorisation. What really defines my location in relation to remote work? Since I’m not bound to one physical location anymore I can split my time between different places. I might move every 6 months, what then?
Still, working remotely hasn't been around for decades, is it all so straightforward? From an EU perspective I can assure you it's a deep mess.
People have a different, long-term relationship with the tax authorities because 'you're not going anywhere' and this makes it possible to settle at a later time. Also tax authorities take action in retrospect specially because things are difficult to prove on-the-spot, but Facebook will need to pay you next month and next year you might be working somewhere else. How will they be able avoid over/underpaying you?