I work for a CA company, but remotely from the midwest (home office), and I pay CA taxes for the stock vests I got from my company last year (because I lived in CA when the grant occurred.) But I also paid taxes in my home state for the same stock. I just filed an "Other State Tax Credit" in CA (form 540 Schedule S), which effectively made CA give me a refund equal to the amount I paid in my home state. Same thing would happen if my regular salary was taxed.
It effectively means you pay the higher tax of the two states: if the state you live in has higher taxes (NY for instance), CA would have to essentially refund you all the taxes you paid; whereas if you live in a cheaper state (more likely), you effectively pay a CA income tax rate overall, since CA is keeping the remainder after refunding you for the other state.
Several of my coworkers who moved from the Bay Area still pay CA taxes for stocks that were granted before they moved, which means potentially 4 years of paying taxes for two separate states.
On the plus side, their CA tax bracket is much lower, since they don't have salary / bonus / new stock grants.
This is especially true if you use an outsourced HR company like TriNet for payroll, which many startups do.
I live in MA and, when I worked in NH, I had to pay MA income taxes because I live here even though NH has no state income tax. Now I work for a company with offices in MA (even though that's not where the official HQ is) so I still pay MA taxes; doesn't matter if I'm remote or not. Were I to move to NH and work remotely, my understanding is I would then not pay state income tax (although I would if I were assigned to an MA office).