Two answers, one short, long; Either way "She and I" sounds better in my opinion, but you would probably kill an English teacher or a grammar nazi just by saying that.
Short Answer: Her and me
Long Answer: English (and most other languages) have subjects and objects in their grammar, where the subject is the thing doing something and an object is the thing being acted on. On a side note, a lot of the languages that have subjects and objects also have different words for pronouns depending on whether they are subjects or objects.
e.g. English I/Me, Latin Ego/Me, Japanese wa-particle/wo-particle, Spanish yo/me
In this example the subject (the car) is being owned by the object (Her and me).
I would say "We own the car together" or "We both own the car - together". "She and I own the car" is grammatically correct, but it feels awkward, and thus would not be used in everyday speech.
The form which you enquire about: "The car is owned by her and me" is grammatically correct thus, but feels strange to me. I might say "The car is owned by both of us together", or more formally "The car is owned by us jointly".
I am British, a native user of English for over 60 years.
"We own the car" but if you want to be absolutely clear, then "We own the car, she and I." "She and I own the car" is correct, but unusual. If you use the passive, then you must use the accusative forms after the preposition: "The car is owned by us", or "The care is owned jointly by her and by me." "By she and I" is what is known as a hypercorrection, i.e. when someone is so concerned to be correct that he (or she) turns what they would say in an informal situation into the exact opposite, and so, not only gets it wrong but also reveals what a pretentious idiot he (or she) is.
"She and I own the car" and "The car is owned by she and I," but try not to use "The car is owned by she and I" in formal writings as it is an passive voice sentence :)
We own the car sounds best. The car is owned by me and her is acceptable. And in Japanese, the "wo-particle" is pronounced and usually transcribed as "o." It is only known as "wo" because in most beginners' hiragana charts, it is on the same line as "wa."