To no one in particular [and by no one, I mean Tracy and Mikelle . . .]
Richard and I’s campsite . . .
Logan and I’s camper . . .
Both of these are nails on a chalkboard for me . . .
I is a personal pronoun that is used only as the subject of a sentence. In your sentence the subject is (probably) campsite or camper, so that eliminates I as the word to use here, since it would not be used as a subject, but as something modifying the subject. Also, I is not a possessive and doesn’t ever get used with an apostrophe (to show possession).
Mine is a possessive personal pronoun that is used only as an object in a sentence. We’ve already determined that the noun here is either campsite or camper, and the word we need is a modifier. Mine isn’t going to work here. It’s a possessive form, but not an adjective, so it can’t modify “campsite or camper”.
My is the word you want. It’s the only one of the three words that is a possessive adjective. You must have an adjective here to modify the noun, campsite or camper, and to work in parallel with Richard or Logan. No apostrophe is needed. You can test this word in a way that might be helpful in getting it to sound right to you by using it in the phrase you have without the other modifier, i.e., Richard’s campsite and my campsite. The correct word should always make sense on its own as well as with other, parallel and correct words. My campsite, and Richard’s and my campsite are both correct.
OK. I can breathe a whole lot easier now.