Are Masters' degrees in Sweden and Denmark free?
I finished my Master's in Sweden in 2009.  I am from The Netherlands.  Higher education is unconditionally free of charge in Sweden.  There used to be a mandatory student union membership with a fee of around 100 SEK/year (a little over €10) but this was abolished by the Reinfeldt government.  It does not matter whether you finish or whether you stay in the country afterwards.

EU does not permit tuition fee discrimination between native and EU students, so as long as higher education is free for Swedes it is free for EU students.  It is no longer free for non-EU students, who in fact pay some of the highest fees in the world.  I believe EEA students are free of charge as well.

However, you still need to finance your own cost of living, which may not be easy as finding a job without knowing Swedish will be difficult.  The Swedish government only provides grants to Swedish students.

To answer your last question, there are plenty of students from other European countries studying in Sweden.