Sort of. For a stay longer than 3 months you need to register (mostly via a Town Hall or Police Station).
As a worker, you'd need your EU passport and a confirmation of employment from your employer if you wanted to stay more than 3 months.
If you were self-employed, you'd need evidence of that.
If a student, you'd need proof of enrolment, and evidence of health insurance.
Similarly a pensioner would need health cover and proof you don't need income support from the new country. The methods of proving these thing can vary from one country to another.
Once you have registered, often you may get a document confirming your right to reside in the new country, generally a registration certificate, in some form, which should be valid indefinitely. Rules vary from one country to another about this, and some countries require that you are always able to produce it on demand with your passport (although theoretically they can fine you in some countries if you don't, they can't deport you).
The actual law (
https://eur-lex.europa.eu/legal-content/EN/TXT/HTML/?uri=LEGISSUM:l33152&from=EN) says:
EU citizens with a valid identity card or passport may:
Enter another EU country without requiring an exit or entry visa. Family members who are not nationals of a EU country are not required either an exit or entry visa if they possess a valid residence card.
Live in another EU country for up to 3 months without any conditions or formalities.
Live in another EU country for longer than 3 months subject to certain conditions, depending on their status in the host country. Those who are employed or self-employed do not need to meet any other conditions. Students and other people not working for payment, such as those in retirement, must have sufficient resources for themselves and their family, so as not to be a burden on the host country’s social assistance system, and comprehensive sickness insurance cover.
Have to register with the relevant authorities if living in the country longer than 3 months.
Their family members, if not EU nationals, are required a residence card valid for 5 years.
Be entitled to permanent residence if they have lived legally in another EU country for a continuous period of 5 years. This also applies to family members.
Have the right to be treated on an equal footing with nationals of the host country. However, host authorities are not obliged to grant benefits to EU citizens not working for payment during the first 3 months of their stay.
In addition
Family members – who are next of kin to the EU citizen – may, under certain conditions have the right to continue to live in the country concerned even if the next of kin themselves dies or leaves the country.
Different rules apply to different group of people - if you are a pensioner the requirements will be different than that for a student or a worker, etc.
You can read all about the requirements here:
https://europa.eu/european-union/life/living-abroad_en