For Finland, you may find information on registering your right of residence here: https://migri.fi/en/registration-of-right-of-residence. Employment is not a requirement to exercise your freedom of movement.
- You have arrived in Finland as an employed person or a self-employed person.
- You have arrived in Finland as a family member.
- You have arrived in Finland as a student.
- If none of these grounds apply to you, you may still stay in Finland as long as you have sufficient funds to support yourself.
For the sufficient funds requirement:
You are not required to have a specific amount of money but you must have enough funds to support yourself. Social assistance may not be your principal means of support. We will take your personal circumstances into account when we consider this income requirement.
EU law is very protective of the freedom of movement and the European Court broadly interprets it in favour of individuals. An EU state cannot arbitrarily deny your freedom of movement and right to settle in any EU state.
Unless you will be dependent on Finnish social assistance during your stay, it is almost impossible for your residence right to be denied on the basis that you are not working and do not otherwise have enough resources. Additionally, as soon as you work or become self-employed, the sufficient resources requirement no longer apply and you have a legal right to live in Finland.
As an EU citizen, you are entitled to the freedom of movement. That means unless you pose a serious danger to society (serious criminality, actively infectious etc.) or are clearly unable to support yourself, your right to settle in any EU country is almost absolute, whether you are going to study, work, retire, or just live there. The EU website has more information here: https://europa.eu/youreurope/citizens/residence/residence-rights/index_en.htm.
A work authorization or permit is not required and cannot be required for an EU citizen under EU law.
However, an EU state may require you register your address with the local government (just like for German and other citizens in Germany) if you are settling for more than 90 days. The other state may require you to prove that you have sufficient financial resources and health insurance if you are neither studying or working in that state.
If your purpose of stay is "jobseeking" and have the appropriate proofs (e.g. communication with companies), you can even stay for up to 6 months without registration or proving your financial resources.