We can say:
She's a friend of mine.
She's a friend of Tom's.
She's a friend of my parents'.
But today I saw this: She's a friend of Jane and Tom.
Is it correct? Or should it be: She's a friend of Jane and Tom's?
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The grammatical form of the sentence you indicate is certainly
She is a friend of Jane and Tom's.
meaning that they both know her and are her friends. However, particularly when speaking, the final "s" happens to be overlooked.
With reference to Jay's answer, the option She's a friend of Jane's and Tom's indicates that they are both friends to this girl/woman, but they do not necessarily know each other.
"She's a friend of Jane and Tom" is correct. The "of" applies to "Jane and Tom" as a compound.
You could also say, "She's Jane's and Tom's friend."
People sometimes say, "She's a friend of Jane's and Tom's." But this is redundant: The "of" already indicates possession; you don't need to also use the "'s".