Renting privately means that the building in which the flat is located is owned by a person (or a company). The rent covers expenses associated with the building as well as the owner’s possible profit. You will have to pay a deposit when you move in.
First, you should search the internet, where a number of private individuals and estate agents have specialised in finding both furnished and unfurnished accommodation on short or long term contracts for expatriate workers and others coming to Denmark. There is a considerable amount of accommodation available if you search under “Housing in Copenhagen.”If you do not want to go through an agency, you can try googling. Use the search word “lejeboliger” (= accommodation to let) and “København” You will find a lot of useful sites. You can find many private advertisements for accommodation to let on Den Blå Avis (in Danish). You may also be lucky if you look in the accommodation section of newspapers or in the Yellow Pages under “Ejendomsselskaber” (Property Companies) or “Ejendomsadministration” (Property Management).