Yes technically you can buy a used car from a private individual and their plate be used if its still valid however most will not let you keep their plate. Too much liablility.
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If you noodle around on Craigslist, you can find beater cars where they say "sorry I lost the title, cash only" or some other such garbage. That would be the guy you'd go to for a completely under the table deal, probably buying a stolen car or who knows what happened to that car. You could also make a paper dealer plate from a template online.