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What happens on a trip stays on a trip.
Literal
Travel [possessive-の] shame [topic-は] scratch-discard.
旅の恥はかきすて is a proverb meaning 'shame committed while traveling can be discarded' — away from home, one's embarrassments don't follow you back. The closest English equivalent is 'what happens in Vegas stays in Vegas.' A culturally revealing proverb about the freedom from social constraints that travel provides.