I don't have much experience with this, but when I rented an apartment with a roommate, we both had to submit proof of employment. Actually, I was added to the lease later, and they still needed proof of employment from me.
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Yeah, most likely both. Landlords like to be able to confirm that each person on the lease is able to sufficiently contribute to the cost of rent, utilities, etc. Turn-over from bad and rent-delinquent tenants is a big problem for a lot of landlords--which can then potentially turn into nasty legal battles (on top of the hassle of trying to recoup the cost of that lost rent, fixing the place up again, showing it, etc.)
Particularly diligent landlords and rental companies will also want to perform a credit and background check. But not always.
Not always. At the last place I stayed at, I was upfront that I couldn't provide proof of income. I really didn't want to sign a lease since my plan was to stay there only temporarily. One of the roommates was moving out and he wanted his name off the lease and my name put in his place. I didn't think it would go through but according the leasing lady, the roommate that was staying had high enough income to rent the whole apartment.
They did do a credit check though. 99% of the time they will do a credit check. Landlords aren't really looking for high credit scores, just don't want anyone with a really bad credit score. And I believe there are also regional lists for landlords of tenants that have been evicted or were sued for not paying rent or for damaging the place. It's like a blacklist. That's what landlords are truly interested in.
Most of the time when I lived with roommates, my name was not on the lease. Landlord didn't even know I existed. The lease holder/roommate never asked for proof of income. It's all very casual. Didn't sign anything. Didn't show any ID. Usually just asked for first month's rent and a deposit. So if the rent was $700 a month, I'd fork over $1,400. Of course, it goes both ways. I had to trust them too. Trust that they weren't going to take my $1,400 and tell me to screw off.