berechtigt
justified
Being well-founded, justified, or legitimate. Often refers to questions, concerns, or criticism.
Deine Kritik ist berechtigt, das hätten wir besser machen müssen.
Your criticism is justified; we should have done that better.

justified / entitled
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justified
Being well-founded, justified, or legitimate. Often refers to questions, concerns, or criticism.
Deine Kritik ist berechtigt, das hätten wir besser machen müssen.
Your criticism is justified; we should have done that better.

'Berechtigt' (justified / entitled) comes from the noun 'das Recht' (the right/law). Something that is 'berechtigt' has a lawful or good justification. A person who is 'berechtigt' has the right to do something.
Das ist eine berechtigte Frage.
That is a justified question.
You can recognize the meaning from the sentence structure. If 'berechtigt' stands alone with a noun (e.g. 'berechtigte Kritik' (justified criticism)), it means 'gerechtfertigt' (justified). If it is followed by 'zu + infinitive', it means 'die Erlaubnis haben' (to have permission).
Ihre Sorge ist durchaus berechtigt.
Her concern is entirely justified.
In the meaning 'die Erlaubnis haben' (to have permission), 'berechtigt sein' is almost always followed by a construction with 'zu'. This can be an infinitive with 'zu' or 'zu + noun'.
Studenten sind berechtigt, die Bibliothek kostenlos zu nutzen.
Students are entitled to use the library for free.
In the meaning 'gerechtfertigt' (justified), 'berechtigt' is often combined with abstract nouns such as 'Frage' (question), 'Kritik' (criticism), 'Zweifel' (doubt), 'Hoffnung' (hope), or 'Sorge' (concern).
Er hatte berechtigte Zweifel am Erfolg des Plans.
He had justified doubts about the success of the plan.
responsible
'zuständig' and 'berechtigt' can both have to do with rules, tasks, or rights. 'zuständig' says who is responsible for a task or area. 'berechtigt' says who has the right or permission to do or receive something. Learners may confuse the words because in both cases a person is officially allowed or expected to do something. The difference is: 'zuständig' asks about responsibility, while 'berechtigt' asks about permission or entitlement.

Eva ist berechtigt, die Unterlagen vor dem Wochenende abzuholen.
Eva is entitled to pick up the documents before the weekend.
Mara ist im Verein für die Termine der Kindergruppe zuständig.
Mara is responsible for the children's group schedule at the club.