Our agreement was that any child conceived by me regardless of the biological father would be the child of myself and MrS. Dude's role - if he were the biological father - would be anywhere from "sperm donor" to "favorite uncle" depending on the state of our relationship. Dude agreed that he was willing to sign a legal document giving up any potential paternal rights should it come to that. The decision as to whether or not to determine paternity, if that became necessary (i.e. child did not look like one more than the other) would have been left to my husband - did he want to know or not? In terms of genetics/medical history - I think knowing the medical history/family history of both is generally sufficient - no major genetic skeletons in the closet on either hand.
I wrote more about our decision/experience in this post: