There are many, many documented reports of people remembering things they could not have known. Either lots of people are conspiring to fabricate these stories all around the world in a really bizarrely coordinated way, or they’re real. There’s a chapter in Soul Searching by R’ Astor about this, but there are probably more thorough and convincing sources available today.
It’s not mandatory to believe in gilgulim, since there are rishonim who spoke against it and it’s not explicitly mentioned in Gemara or Tanach. But virtually every major rabbinical authority since kabbalah was popularized 500-1000 years ago has believed in it (if they said anything about it). The Rashash is the only exception I know of.