"The "Modern Kindergarten Teacher" is a local newspaper."
Translation:A "Modern Óvónő" egy helyi újság.
If it is part of the name, then it is part of the name. if it is not, then it is not. Then the quotes mark what is in the title. Hungarian newspapers usually do not have a definite article as part of the title, unlike English newspapers. But that is correct, the sentence may still need one for grammatical reasons.
If the title happens to have a definite article, then it becomes a little bit ambiguous, as grammar would still prefer a definite article but it would be weird to have two of them next to each other.
Above, the title of the newspaper is "Modern Óvónő".
If the title were "A modern óvónő", the grammar of the sentence would still want an article:
Az "A modern óvónő" egy helyi újság.
But that is just too weird. The outside article is usually just omitted.
"A modern óvónő" egy helyi újság.