The stock market has made it easier than ever to see what famous investors, executives, celebrities, and politicians are buying. A regulatory filing, social-media post, or headline can put a trade in front of millions of investors before the ink is dry. That access creates the illusion that copying a successful investor can shortcut the hardest part of investing: doing the work yourself.

MarketWise conducted a survey of 1,005 U.S. retail investors in July that shows just how widespread the practice has become. Half of respondents said they had copied a publicly disclosed trade by a famous person. But the same survey also reveals the problem: Copying someone else’s trade can turn investing into a game of telephone, where the last…