If you use puts, that is a tensed verb, and has to be interpreted as parallel with might:
which [might result in them losing money] and [puts their privacy at risk].
so it is stating that it does put their privacy at risk. This is probably not what you intended.
If you use put, then it is not a tensed verb, so it must be inside the scope of "might":
which might [[result in them losing money] and [put their privacy at risk]]
This distinction only appears in the 3rd person singular ("he/she/it puts"). In any other person, there would be no distinction, and the sentence would be formally ambiguous (eg those things might result in them losing money and put their privacy at risk. ) Only context (and real world knowledge) can then resolve the ambiguity.