It really depends on the country, the EU is not quite as homogeneous at it seems from the outside.

Typically, master students are expected to write a master thesis, and that often is research-focused. Often, they're supervised in a daily fashion but a PhD student with a professor being responsible. This happens in the labs, but because every student goes through it, these students are not always listed as lab members (because they aren't).

Sometimes, labs also have funding for student researchers. They do work that's not part of their thesis, and that's much rarer, and in most countries, the pay is shit compared to jobs in industry (I guess: welcome to academia).

If you want to get published and don't happen to find a lab that's hiring students: Choose a good thesis topic where the lab is 1) doing research 2) you contribute to an ongoing effort in the lab 3) you are supervised by someone who wants you to succeed (i.e. your project isn't a side-project, or very specific nieche follow-up), and 4) you make your ambitions clear from the beginning.
Don't expect the supervisors to give you an idea that end-to-end will lead to a publication. You need to use your own judgement, and your research on the field, to make a somewhat educated guess.

Unless you're trying to join Ivy League like places where currently a whole pipeline of FAANG engineers pushes their high-school students as interns through a pipeline to have their names top-tier conference publications before they even graduate high school, you'll end up being a decent candidate with:

• good enough grades
• a decent selection of courses that gives you solid background to understand enough math and theory
• side-projects, interesting term papers, ba/ma thesis projects
• network, get to know your lecturers, engage with them
• recommendation letters are often a requirement. Just having good grades doesn't make you a good candidate for a letter. Wtf is the professor going to write? "He was good in my class" is just a weak recommendation letter, barely better than none at all.
• try to particpiate in summer schools, there often is funding for it. typically, summer schools target phd candidates, but you can join them as a master student as well
• if you're rich/well-funded enough, you can even attend academic conferences on your own/as a student
• don't wait for research student jobs to be advertised. Go ask. That's how they're all gone before they make it to the job boards.

Probably more, but I can't think of more rn.