If you are an EU citizen and if you are under 18 for example if you are living somewhere close to the border to another EU country can you enter the other country without any letters or permits by only holding your ID card?
You are covered by the EU freedom of movement, so you certainly have the right to travel with your ID card, without any passport, visa or permit.
Beyond that, border guards or the police could be concerned about your safety as a minor and try to ensure that you have not run away or been abducted. They might therefore run your name through some databases of missing persons/persons to watch for their own safety (those do exist, beside lists of entry bans, wanted persons and stolen documents).
That's also why it's usually strongly recommended to have some letter from your parents or legal guardian, which could also be translated or notarized, depending on the country. Some countries also have or had some standard form/document to that effect but in most places I don't think that it is mandatory. It's just easier to avoid any discussion.
The way you will be treated also depend a lot on the specifics (how old you are and what you are doing). If you are a teenager, can show you live in the area and take a local train or bus across the border to go shopping with friends, I assume nobody will bat an eye. On the other hand, if you were a young child looking distressed and you were leaving for a far-away country with only one of your parents, this would raise concerns that this parent is trying to take you away during a custody dispute or something like that.
Note that unlike visa/permit issues, which are mostly about preventing people from entering, these issues often come up on exit. For example, until 2013, France had something called an “authorization to leave the territory” that parents could request to let their minor children leave the country alone.
But in many cases (e.g. in the Schengen area), there won't be any systematic check at the border so none of this matters much. You definitely have the right to be present in another EU country with your ID card (i.e. even if you could in theory be stopped from leaving the country, you certainly cannot be fined or otherwise punished because you did it anyway).