Disclaimer: I do not claim to have any expertise in any shape or form pertaining to European graduate admissions processes. I am very nervous about this part because I know I WILL be questioned about them when I will give my student conference on the topic of graduate school admissions abroad (from a Canadian perspective). I will admit that I am biased towards one destination because I applied to 12 schools in the US (a list that might come across as wanting to redeem from a broken undergraduate dream but, in the context of physics, actually represents a broad range of schools)…
Of course, at my department, there are some of these self-righteous students who would outright refuse to leave their home region. There are also some students who would not consider graduate study in the US because of tests (GRE, both general and subject, where applicable, TOEFL) but who would readily consider graduate school even though the logistical barrier (and, in some cases, language barrier) is higher. Others also quote the unappealing local culture of some destinations although one’s ability to immerse in the local culture is necessarily limited in graduate school vis-à-vis undergraduate study-abroad, due to the intense workload of graduate school, especially a PhD.
However, one caution one should take is that fully-funded masters in the Canadian sense are usually not sensible in continental Europe because, despite the large variation between countries, and sometimes within countries, in Europe, masters are more coursework-heavy, and you do little to no research since Europe masters are structured more or less like 2 more years of undergrad and hence charge tuition (nominal outside of UK) like undergrads. As a result, doctorates have little or no coursework. It is a result of the chasm happening between masters and doctorate there, instead of between bachelors and masters (or bachelors and doctorate in the US) as could be the case elsewhere in the world.
One thing that is quite different is that PhD admissions in Europe tend to ask for an outline of your proposed dissertation and that you have, depending on the country and university, 3, 4 or 5 years to defend. And that one holds a masters, in which case masters grades enter the equation to a greater extent than in the US. But, since test scores are not a factor, the application-independent expenditures (e.g. expenses that you incur once regardless of the number of applications submitted) are limited to the transcript translations (whenever applicable) or the student visa (if admitted).
If you have any additional information, feel free to comment with that information at hand.