When it comes to the beggars in Sweden, they're generally not Swedish citizens, and as such I don't think they qualify for the housing assistance programmes we have here. There definitely are homeless people in Sweden, according to this report from Socialstyrelsen there's more than 27000 homeless people in Sweden.

There are a couple of definitions of homeless here though.

Situation 1: Acute Homelessness
Individuals sleeping outdoors or in stairwells, cars, tents, etc., and also those staying in emergency shelters, hostels, or similar temporary accommodations.

Situation 2: Institutional Stay and Supported Housing
Pertains to individuals in prison, healthcare institutions, or supported living who will leave within three months but do not have a home to go to.

Situation 3: Long-term Housing Solutions
Refers to living in special housing solutions provided by social services, where the accommodation is linked with supervision and specific conditions or rules, like trial apartments, training apartments, and social contracts.

Situation 4: Self-arranged Short-term Housing
Living temporarily without a contract with friends/acquaintances, family/relatives, or under a temporary lodging or subletting arrangement with a private individual.

This page on Boverket claims that in 2017 there were 32400 homeless people, 18% of those (5900) were classified as acutely homeless.