What is a Muslim's duty upon hearing a Hadith and doubting its authenticity, especially if heard in a public place where it is difficult to consult the speaker?

Muslims must verify and ascertain the authenticity of any hadith before attributing it to the Prophet, peace and blessings be upon him, for he said: "Lying about me is not like lying about anyone else. Whoever lies about me intentionally, let him take his seat in Hellfire." And whoever knows that a hadith attributed to the Messenger of Allah, peace and blessings be upon him, is false, must inform the one speaking it; for this is part of sincere advice to Allah, His Messenger, and the Muslims. Sincere advice to the Messenger means reviving his Sunnah, spreading his Sharia, refuting accusations against it, and refraining from speaking about it without knowledge.