There have been of cases in which a hospital has been very suspicious that perhaps money is being secretly (and illegally) paid to the stranger in return for the "donation", so they may investigate more carefully. I don't know if there are any hospitals that automatically exclude internet solicited donors these days, as there have been so many genuine donations from strangers.
But keep in mind that there are a lot of things that a patient can do if he/she does not have a living donor:
--- if there is a relative, friend, neighbor, etc. who is willing to donate but is not a compatible blood type, sometimes a "swap" or "exchange" can be arranged among other patients in the same situation.
www.kidneyregistry.org has a large data base and tries to make some matches.
--- the patient should make sure that relatives, friends, neighbors, co-workers, alumni, church/synagogue members, etc. are aware that a kidney is needed, and that living donation is an option. Sometimes once the need is KNOWN a volunteer comes forward from among these groups (either to donate a kidney directly, or through a "swap")
I hope this information is of some help.
best wishes,
Fr. Pat