Learn Farsi: این ماشین یا عمومی است یا شخصی.

They both use the suffix -i to turn a noun into a relational adjective (an adjective that means related to that noun). The word omumi comes from the noun omum (the general public), so adding -i makes it mean public. Similarly, shakhsi comes from the noun shakhs (person), so it means personal or private.

I see the word ya used twice here. Is that how you say either... or in Farsi?

Yes! In Farsi, you just repeat the word for or (ya) to create an either... or pair. So, ya omumi... ya shakhsi literally translates to or public... or private, which is the standard way Farsi expresses either public or private.

Why is the verb ast (is) in the middle of the sentence instead of at the very end?

Normally, verbs go at the very end of a Farsi sentence. However, in an either... or (ya... ya...) structure, it is common to place the verb right after the first option. You could say in mashin ya omumi ya shakhsi ast, but breaking it up as in mashin ya omumi ast ya shakhsi sounds very natural and balances the two choices.

Why is there no connecting ezafe (the -e sound) after mashin?

Because omumi (public) is not directly describing the car in a noun phrase. If we wanted to say this public car, we would use the ezafe: in mashin-e omumi. But here, we are making a full statement: This car is public. The ezafe links a noun directly to an adjective, but we do not use it when they are separated by a verb.