the child lives with his father and his mother lives on the east coast, but comes to visit him monthly.
Were the parents ever married? If not, was paternity ever established, and is there a court order regarding custody, visitation and support? If they were married, are they now divorced? In what state was any divorce judgment or order entered?
The parents have joined custody
As you've described the living situation, that cannot be true. They might have joint (not "joined") legal custody (which relates to decisions regarding schooling, religion, medical care, etc.), but they clearly don't have joint physical custody.
If he moves in with us and God forbid something happens to the child at a party or whatever... can the parents come after us stating "we didn't allow" the kid to go to a party etc.?
Can they "come after" you? I don't really know what that means, but the answer is probably yes. If you're asking about legal liability, that's fact-specific and can't really be answered in the abstract.
what rights would we have?
If you agree to allow their child to live with you, you would not acquire more rights as a result. You would be undertaking responsibilities and would owe legal duties to both the child and his parents. The parents could (and probably should) give you certain authority to act in their place (either by signing a power of attorney or by seeking a guardianship through the court).