What's the grammatically correct way to indicate shared possession: "John and I's car", "John's and my car" or something else?

The second is correct. It would also be correct to say, "our car".

In this situation, you figure out how you would say it if there was only one owner, and then combine those forms of possession together. For example, you would say "John's car... but you would (hopefully) never say, "I's car"; you would say, "my car". Therefore, the correct form is "John's and my car". You would determine whether to use the possessive pronoun or the actual name depending on who is speaking/writing, the identity of the audience (the person(s) to whom you are writing/speaking), and whether or not the audience is already aware of the identities involved.

If the car belonged to John and his sister Mary, it would be "John's and Mary's car" or John's and his sister's car" or "His and Mary's car."

If the car belonged to you and your husband John, it would be "Your and John's car" or "John's and your car" or "Your and your husband's car".

If the car belonged to you and your sister, and also to your father, it would be "Our and our dad's car."

If the car belonged to John, his sister Mary, and their father, it would be "John's, Mary's, and their dad's car" or "Their and their dad's car" or, simply "their car".

You get the idea...

John's and my car. Or, the car that John and I own.

There is no such possessive pronouns as "I's".

' Our car ' your suggestions are clumsy reading , cluttered . ' John's and my car ' was your best offering , probably correct too .

Mine & John's car or just our car.