If I am correct
I have allowed smth would be rendered in German as Ich habe etwas gedurft
and
I have been allowed smth would be rendered as Ich bin etwas gedurft worden.
Are these correct? If not, I'm always willing to improve myself.
The first is incorrect, because the meaning changes. You are allowing something, but your translation means that you have been allowed for something, this would be:
Ich habe etwas erlaubt
The second is incorrect too (sentence structure), this should be something like that:
Ich habe etwas machen dürfen
Ich habe etwas gedurft (used rarely)
Mir wurde etwas erlaubt
Ich habe etwas machen dürfen
dürfen
is not implicitly passive. You just translated it by a passive construction in English. Correct would be: dürfen
is a modal verb, and as such behaves differently than usual verbs. Most important aspect here: In spoken or casual language, modal verbs are rather used in Präteritum than Perfekt. That's why indoxica's propositions sound odd.