When sneezing we are told to say hamdalah. Rasulullah Shallallahu’alaihi Wasallam said:
إِذَا عَطَسَ أَحَدُكُمْ فَلْيَقُلْ : الْحَمْدُ لِلَّهِ ، وَلْيَقُلْ لَهُ أَخُوهُ أَوْ صَاحِبُهُ : يَرْحَمُكَ اللَّهُ
“If one of you sneezes then say ‘Alhamdulillah’, and his brother or friend (who hears it) should say:” Yarhamukallah “” (Narrated by Al Bukhari).
From this hadith we also know that the hamdalah is said with jahr (loud), not sirr (vague), so that it can be heard by those around us. If not, then there are no rules for people to be ordered to tasymit (pronounce ar yarhamukallah), because they are told to do tasymit if they hear hamdalah, as in the hadith:
إِنَّ اللَّهَ يُحِبُّ الْعُطَاسَ ، وَيَكْرَهُ التَّثَاؤُبَ ، فَإِذَا عَطَسَ فَحَمِدَ اللَّهَ فَحَقٌّ عَلَى كُلِّ مُسْلِمٍ سَمِعَهُ أَنْ يُشَمِّتَهُ
“Allah likes sneezing, and does not like tasa’ub (yawn). If someone sneezes then say hamdalah, and it is his right to every Muslim who hears it to tasymit “(Narrated by Al Bukhari).
Imam Malik said:
إذَا لَمْ يَسْمَعْ حَمْدَ الْعَاطِسِ فَلَا يُشَمِّتُهُ إلَّا أَنْ يَرَى تَشْمِيتَ النَّاسِ لَهُ فَيُشَمِّتُهُ
“If someone does not hear hamdalah from someone who sneezes, then he does not need to be tasymit, unless he sees other people tasymit to him then he also participates in tasymit” (Al Fawakih Ad Diwani, 8/416).
Once again, you should say hamdalah when sneezing with jahr, not vague. Ibn Abidin stated:
وَيَنْبَغِي لِلْعَاطِسِ أَنْ يَرْفَعَ صَوْتَهُ بِالتَّحْمِيدِ ، حَتَّى يُسْمِعَ مَنْ عِنْدَهُ فَيُشَمِّتَهُ
“The person who sneezes raises his voice in saying hamdalah, so that it is heard by the person next to him, then he is tasymitic” (Raddul Muhtar, 27/59).