Paperwork and deposits for apartments are reasonably similar to what's required in the US though the standards seem to favor renters. I don't know if it's more or less similar in other European countries (or elsewhere).
Citizenship is not required. Of course, there may be some racial discrimination in practice but it's possible to file a complaint. For example, making a non-citizen pay a special deposit was used as a specific example of indirect discrimination (I couldn't find an English translation for the linked document).
Besides normal rental apartments, many people live in "housing cooperatives" (bostadrätt). These are somewhat similar to condominiums in practice but different in aspects of ownership. Skipping the details, the reason to mention them here is that it's often possible to rent these "second-hand".