It's optional: you can drop pronouns in Romanian. Examples:
(Eu) te cunosc (pe tine).
(Tu) îmi ești prieten (mie).
(Eu) îți sunt dator (ție).
(El) o cunoaște (pe ea).
This applies to all personal pronouns in the nominative, accusative and dative cases. You only really need to use them when you want emphasise something:
Nu pe el îl iubesc, ci pe tine! - I don't love him, I love you!
Nu mie trebuie să-mi dai bani, ci lui! - You don't have to give money to me, you have to give it to him!
But why are there two forms of the accusative pronoun in the first place? In your examples the first pronoun is dropped because it is indicated by the verb, but 'te' and 'pe tine' mean the same thing.
Yes, you are right. The only one you really use is the first one (which is mandatory). Forms such as „pe tine” or „mie” are really only ever used in order to emphasise something and you never hear or see them otherwise.
As to why that is, I must admit I have no idea, but that's how it is.
Romanian isn't alone in this regard, as at least one other Romance language shares this little quirk. In Portuguese this sentence would be "A Maria não te conhece a ti", where a ti serves the same purpose as pe tine in Romanian - reinforcement - and is also entirely optional and seldom used in colloquial speech.
te and pe tine is totally different for my native language (greek) because romanian and greek have similar way of expressions in some ways. I believe that people whose native language is english cannot really understand that because it is easier and more simple language example: (Eu) te cunosc (pe tine) (Εγώ) σε γνωρίζω (εσένα) I know you both languages have the ability to emphasize when it is needed
When speaking the English sentence, you can change the meaning by emphasing different words. e.g. "Does not know YOU" is making the point that it is just you that is not known. Romanians have this technique available in their written language.
i think these sentences are basically meant to teach us the phonomenon of redondancy in romanian : the direct object of a verb is normally anounced by a pronoun that in other languages is only used to replace the object.