As said already there is only real demand after gaining a couple years post graduation experience. Competition is higher than ever before for entry level graduate roles, as evidenced by average graduate wages stagnating/decreasing in real terms- rising unemployment and underemployment stats.
So I would do a lot of research before choosing engineering nowadays. You need to really know what you're doing.
I'd say for some roles you need to have part-way trained yourself in order to be close to minimum standard, or have substantial experience in a particular area through placements/ but it varies a lot (there are tens of thousands of graduates and jobs each year across Engineering and Technology).
Higher demand for experienced engineers could be due to people leaving the profession for better pastures (aka any other management/generic graduate role) or simply fewer entry level roles than is needed to maintain an aging workforce.
It is a similar situation in other countries e.g. Canada/Australia (high ratios of graduates to entry level jobs despite better engineering industries) so not simply a case of opting to move abroad. It is a bit misleading the way university prospectuses "advertise" engineering as being in demand worldwide as again that comes later.