Nationals of other countries do need a Danish work permit before they can take up employment. Work permits in Denmark should be applied for roughly three months before arriving in the country.
You do not need a work permit to take on employment in Denmark if you hold a residence permit through family reunification, asylum or on humanitarian grounds.
Obtaining a Danish work permit is entirely your responsibility. Working illegally in Denmark could lead to deportation as well as either yourself or your employer facing a fine or a prison sentence. These rules also apply for voluntary and unpaid work.
The Danish authorities will make a decision about your work permit based on whether there are sufficiently qualified people already living in Denmark who could take on the job you are applying for. They will also decide whether or not the skills required for the job are specialist enough to deserve a work permit.
Whatever happens, you must have a written contract of employment or job offer that outlines your salary and employment conditions, both of which must be in line with Danish standards.