The Niskanen Center has published a new position paper on “Private Refugee Resettlement in the U.S.: History”, link here. The paper is by David Bier and Matthew La Corte.
The paper proposes that private charities (which may be faith-based) could raise funds for resettlement, funding people when funds are available.
There is discussion of the Personal Responsibility and Work Opportunity Act (PRWOA) of 1996 which sometimes means that entry of an alien immigrant means that a relative or other private individual(s) must promise to provide personal support. This obviously creates moral dilemmas around the expectations that individuals should become socialized to provide “radical hospitality” -- the whole naive "spare bedrooms" idea.
A few states (at least South Carolina) want to force sponsors to indemnify the public against any criminal behavior by immigrants, which is a moral burden not many individuals would take. The effectiveness of the vetting program is an issue, as it is very difficult or Homeland Security to do this in war zones like Syria. The GOP idea of helping other countries set up major safe haven spaces, with military protection, sounds wise to me.