How do prions multiply without nucleic acids?

Structure change. Prions are abnormally folded versions of normal proteins. When a prion comes into contact with the "normal" version of itself, it changes the normal protein's structure into a prion.
Low energy. Amino acid strings can fold in a gazillion ways. Usually the biological config is the lowest energy one. Prions are protein fragments that have a bio function that is perhaps not at the lowest energy. This opens the door to versions of the same string folding "more deeply". As a few such are energetically favorable, they recruit surrounding hio material to the same abnormal config, zombie-like.
More than this? If answers you've received to this question are enough, fine. There is a lot of information on the Web about prion diseases. If you'd like to hear it from the horse's mouth, go to YouTube and search for talks by Stanley Prusiner, M.D. He won the Nobel Prize in Medicine for the work he did on prions, which laid the foundation of what we know about them today. (A Jew, I might add.)