It means 'collected in their chests' i.e. learned and memorized. In the same sense that the Quran says:
لا تحرك به لسانك لتعجل به
إن علينا جمعه وقرآنه
Move not your tongue with it, [O Muhammad], to hasten with recitation of the Qur'an.
Indeed, upon Us is its collection [in your heart] and [to make possible] its recitation.
— Quran 75:16-17
{إن علينا جمعه وقرآنه} أن نجمعه في صدرك
'It is for Us to collect it' means we will collect it in your chest
— Bukhari 4928 and Muslim 448a
And there are many other examples of texts in ahadith and athar of the sahaba where 'collecting the Quran' is used to refer to one who is a Hafiz, such as Bukhari 5036, Abu Dawud 587, Ibn Majah 1346, Jami at-Tirmidhi 2382 etc.
At least some of these Sahaba were also scribes and wrote down verses of the Quran on physical objects during the lifetime of the Prophet ﷺ. However, I am not aware of any evidence that they compiled a bound paper mushaf at that point, rather it is known that it was done later during the caliphate of Abu Bakr.
قوله جمع القرآن أي استظهره حفظا
— Fath al-Bari