How do spouses living in different states file taxes?

You will generally never have to pay state income taxes twice on the same income. You might be taxed by two states for some portion of your income, but in that case, you would be able to claim a tax credit on one state's income tax return for the tax paid to the other state on that piece of income (up to the amount of tax you paid for it in the first state), so the net effect is you pay whichever is the higher rate of tax between the two states on doubly-taxed income.

Generally, residents of a state are taxed by that state on their worldwide income, while nonresidents of a state are taxed by that state on only their income from that state. You are a resident of CA, and a nonresident of DC, while your future spouse wife will be a nonresident of CA, and a resident of DC (or at least she will be after her move; for 2020, she will be a part-year resident of CA (resident before her move) and part-year resident of DC (resident after her move); she will have to split the tax treatment accordingly). So CA would tax your worldwide income and your wife's CA income, and DC would tax your DC income and your wife's worldwide income.

However, another complication is the fact that California is a community-property state. That means, since you are domiciled in California, your income after marriage is considered community income, and so half of it is considered your income and half of it is considered your wife's income. The half that is considered your wife's will be CA-sourced income of a DC resident, so it will be taxed by both CA (since it's CA-sourced income) and DC (since it's income of a DC resident). So this basically means that all of your income will be taxed by CA, and half of it will be taxed again by DC, and you will have to claim a tax credit for that half that is doubly-taxed. In the case of CA and DC (it may be different for other pairs of states), the tax credit for doubly-taxed income needs to be claimed in the person's state of residence, so in your case, on the DC tax return, since it's CA income of a DC resident (your wife).

All of the above is true no matter if you file jointly or separately. Filing jointly doesn't mean that all of both of your incomes will be taxed by both states. When one resident and one nonresident (or part-year resident) file jointly, they are usually supposed to use the appropriate nonresident or part-year resident form so that only part of the income is taxed. For CA, a joint filing of a resident and nonresident uses the 540NR nonresident form, and taxes will be based on the portion of your joint income that is taxed in CA, and the rate of tax based on your whole joint income. For DC, I believe you would have to file separately if one is a resident and the other nonresident, but it is better for married couples to file separately in DC in general anyway (since DC has the same brackets for filing jointly and separately); your wife would claim the out-of-state tax credit on her separate DC return.