I'm absolutely amazed at the tone and content of some of the responses here. There seems to be a simmering intergenerational resentment that I find hard to fathom. The way I look at it, building wealth is a multi-generational family project. We have, or should have, evolved past the "push them out of the nest" strategy of the animal kingdom! Each succeeding generation should aspire to leave the overall family situation in a healthier financial state than they were born into themselves.
To me, this is a fundamental parental responsibility. It's your choice to have them, after all, so do right by them! First comes education in basic values of self reliance and responsibility, so teach them to cook, clean, shop, tidy, do laundry and basic household, garden and car maintenance. That goes for boys AND girls, BTW.
Then comes education in financial matters and the instilling of good habits. Explain how the family finances work, and what goals the family is working towards. This includes avoiding/minimizing short term debt, financing the family home and making provision for parental pensions, children's education and other investments.
Then it's education and getting them on a good career and financial trajectory. Make it clear that you will help to the extent you can to get them set up and established but that it's ultimately their own responsibility.
The key thing here is balance. You shouldn't go beyond what you can reasonably afford. And parents need to look after their own essential needs like mortgage and pension provision, before helping out adult children.
Applying these principles, I would suggest that the "rent" question solves itself. As long as they want to stay at home, that's fine, they are welcome, it's the family home after all. .It should come fairly naturally that they help out a bit at home, both by doing a fair share of the household work (which they've learned to do anyway) and by picking up expenses like groceries, the odd bill, or perhaps buy some furniture or other stuff for the house.
Encourage them to save to buy a place of their own. Matching their savings, euro for euro, is a good idea if you can afford it. (Again, parental assistance should not be at the expense of necessary parental pension provision, or - worse still - equity release from the family home.) If they do want to move out and rent, crunch the numbers with them and show how that eats into savings potential and lengthens the time before they can buy. Then it's their choice.
And instil the idea that what you're now doing for them, should hopefully allow them to do likewise for their children (your grandchildren) when that time comes.
Finally, tell them what provisions you've made in your wills and give them an idea of what they might expect to get - hopefully in many decades time!