I have found this explanation:
The one who leaves, says “Allahaısmarladık” or “Hoşça kalın”.
The one who stays behind, says “Güle güle”
“Allahaısmarladık” is literally, “I leave you to God’s care”. It is almost the same with “Goodbye” “Goodbye” in English is the clipping of “ God bless you”.
“Hoşça kalın” corresponds “Stay well” in English.
“Güle güle” corresponds “Go well” in English.
Either “Allahaısmarladık” or “Hoşça kalın” is correct in informal situations. But in formal situatiaons “Hoşça kalın” is better.
Rica ederim- You're welcome or Bir şey değil- Not at all! - you'll here in the same use in many languages.