If you decide to record or license your own version of a public domain work, you wouldn't need to secure a mechanical license or pay royalties, unless you are using a copyrighted arrangement of that song.

Holiday music is often an area where many questions arise. Many classic Christmas songs that are presumed to be in the public domain are actually copyrighted, so make sure to double-check your sources before just assuming a track is in the public domain. 

A good starting point is the website PD Info (www.pdinfo.com)  if the liner notes and copyright information are not available. You can also search The ASCAP repertory (www.ascap.com/ace). The repertory lists valuable contact details that will help you determine whether a work is protected or not.