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!
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.
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