The first answer to questions regarding taxes and creating a business structure is to consult tax and legal professionals.
I don't know about Germany, but you mentioned that you are interested in other countries as well, so I will answer this for the United States.
Again, you should consult the above mentioned professionals no matter which country you are in.
In the U.S. you are automatically a business when you make money which isn't already accounted for by some other tax scheme. In other words, the money isn't from your employer (which is a W2 form in the U.S.) and it's not counted as income from another business you own. In the U.S. that makes you a sole proprietor. There is nothing special that you need to do for having your own sole proprietorship. It's automatic.
I imagine Germany is similar because there needs to be a tax form to deal with that income. There is nothing blocking payments just because you didn't formally setup a business structure. You got paid, now you need to pay your taxes. There will be a way for Germany to get its share.
Additionally, there is likely some local things that you may need to do. Generally the city requires that you get a business license or something similar. That has nothing to do with your business structure though. There also might be laws regarding where you are able to work from. Generally these laws are in place to make sure you aren't setting up a factory in a residential area. How many people in your situation actually bother getting their businesses licensed is another question.
Another item to note is that in some cases, income from abroad is taxed differently from income which comes in from within your country. For example, in the Philippines (where I'm currently living) there are special economic zones in which you get a tax holiday for all income from abroad while operating from those zones. Other countries have similar.
Of course, I'm not a tax or legal professional. Do your own research. If you feel that consulting pro's is too much for the income that you expect to receive, then I imagine Germany has a website where you can get more info. Download the forms. read the info. I'm sure you will see boxes on the forms you typically fill out for income which is for everything that's not your employer.