Yes and no... The OP doesn't mention if they have kids yet. If they don't, they might not need life insurance unless they have large expenses that the surviving spouse wouldn't be able to cover with just their salary (the first comes to mind is mortgage?). But assuming that you are planning to still live relatively modestly as you start your attending careers, I think it's pretty safe to say that individually, you each make enough self-insure yourselves.
Once kids come into the picture, I'm not sure that I agree with flipmd. The surviving spouse will presumably still be paying off their own student loans (the spouse that passed away should have their loans forgiven) as well as trying to save for their kids' college tuition, pay for additional childcare/household help (since they'll need it if they're a single parent, although the amount needed will depend on their kids' age), save for their own retirement, all while being taxed at a fairly high rate because of their income bracket. As others have said, life insurance is relatively cheap especially if you're young & healthy. I think if you have kids, it generally makes a lot of sense to put a little money into insuring yourselves for the "just in case" scenario.
That said, I also agree that disability insurance is crucial. It's not nearly as cheap as life insurance, but others have already pointed out a number of reasons that it could be financially devastating for one spouse to become disabled... I think regardless of whether you have children, you should probably get disability insurance.
My 2 cents 🙂