Sweden is a member of the EU and any EU citizen has the right to travel to, reside and work and reside in any other EU country. (There are a few restrictions temporarily applying to some of the newer EU states but they're not relevant here).
Any EU country seeking to impose restrictions on travel, by EU citizens, to other EU countries would be in breach of EU law. (Once again, there are a few isolated exceptions to that rule, mainly to do with travel by football fans without tickets but, once again, they're not relevant here).
Swedish citizens leave the country every day without restriction. e.g. many of the residents of Malmo, in Sweden, cross the Oresund Bridge daily, to commute to work in Copenhagen in Denmark. There are no border formalities of any type and definitely nobody demanding to see 1000 dollars before they can go to work!
Lastly, if the Danish government did levy any charges on its citizens, for any reason, the amount quoted would be in Euros and not in dollars!
All of which goes to prove Essex Bloke's concise and accurate summary of the situation: It's a con!
Chris