Forming a noun-phrase sounds best to me: "Tom is Jim and I's friend." You are forming a posessive noun phrase, which includes yourself. You aren't supposed to put yourself before others. So "Me and Jill... " is wrong, amd you're supposed to say "Jill and I..." http://english.stackexchange.com/questions/4226/my-wife-and-is-seafood-collaboration-dinner

"Tom is Jim and my friend" --> Though this seems like it would work, it comes out confusing. It would be better to say "Tom is both Jim's and my friend" to show that Jim is in possession of Tom's friendship. As it is, this sentence would be confusing. "Tom is my and Jim's friend" This sounds awkward. It would sound better to say "Tom is my friend and Jim's friend" "Tom is Jim's and my friend" This works! Other ways to say the same sentence would be: "Tom is both my friend and Jim's [friend]" Friend is implied and does not need to be included, unless you want additional clarity. "Tom is a friend to both Jim and I" "Jim and I have a friend named Tom" But both of these sentences avoid the possessive. This website may help also: http://english.stackexchange.com/questions/86993/apostrophe-in-multiple-plural-posession

Here is my answer, which I incorrectly added above as a comment: As you can tell, native English speakers have trouble with this. Common, casual American speech would avoid the problem altogether, and say: "Tom is friends with Jim and me." The grammatically correct construction is: "Tom is Jim's and my friend." The handy practice aid is to state the question without the intervening name, and just use the pronoun: "Tom is . . . my friend". Or "Tom is friends with . . . me." Then simply insert the other name in the sentence first.

"Tom is mine and Jim's friend" would be the most normal way to say it in conversation.

Sunshine's comment is what would most likely be encountered, exactly because the grammatical construction seems awkward. Native speakers would likely do as she has suggested and work around needing to use the possessive form. Vanessa also suggested the same idea - to not use the form at all.