There is a דין [1] that the 10 sons of המן [2] have to be said all in one breath, to show that they were all killed and hanged in one shot. The source of this דין is רב אדא from יפו, who says that one must say the names of המן's 10 sons and עשרת (the following word) in one breath, to show that they all died the same moment as each other.
מהרי Segal, however, says that one should begin from חמש מאות איש in the previous פסוק because the 10 sons of המן were שרי חמשים over those 500 people [4]. דרכי משה wonders why, if, after all, the 'גמ only says the 10 sons of המן specifically. -?!- It must be that those who do so hold that one should not pause for breath between פסוקים [5].
Now, while רב אדא does not specify by whom it is that the 10 sons of המן should be said in one breath, the רמבם [6] specifies that the intended subject is the Reader. Though the רמבם is not specifically cited, nonetheless חיי אדם [7] references the מנהג of some places that everyone says the 10 sons of המן aloud in one breath, writing that this is not a (legitimate) מנהג.
ערוה"ש, however, considers the 10 sons of המן to be in the same category as other פסוקי גאולה that the congregation says aloud, in addition to the Reader [8]. Another reason argued by the [9] Rogatchover is that שומע כעונה, meaning that the congregation fulfills their duty of קריאת המגילה by listening to the Reader, only works with regards to the actual reading itself. Fulfillment of a specific way of reading, however, cannot be done by means of שומע כעונה. To that end, the Rogatchover would hold that the congregation actually must say the 10 sons of המן aloud in order to fulfill their obligation of saying them in one breath.
[1] שו"ע או"ח תרצ:טו
[2] אסתר ט:ז-י
[3] :מגילה טז
'תוס there specifies that בדיעבד one is יוצא even without doing so.
[4] מהריל הלכות פורים אות יג
[5] ועיין בב"י שם טור או"ח שם ס"ק ג
[6] מגילה ב:יב
[7] קנה:כב וכך הובא להלכה במ"ב ס"ק נב ובכה"ח שם אות צא
[8] או"ח שם סעיף כג
[9] שו"ת צופנת פענח השמטות להל' גירושין חלק ג עמ' 113