He recited quietly at the beginning of his prayer, then someone else joined him; should he complete it reciting quietly or out loud?

Reciting out loud in prayer is a Sunnah for the imam. As for the one who
is praying on his own, he has the choice of reciting out loud or quietly. 

Ibn Qudaamah (may Allah have mercy on him) said:

Reciting out loud is prescribed for the imam, and it is not prescribed for
the one who is praying behind an imam, and there is no scholarly difference
of opinion on this point. That is because the one who is praying behind an
imam is enjoined to listen attentively to the imam. In fact he is not
allowed to recite, so that he may do that (listen attentively). In the case
of one who is praying on his own, what may be understood from the reports
narrated from Ahmad is that he has the choice. The same applies to one
missed part of the prayer with the imam, so he gets up to make up what he
missed. Al-Athram said: I said to Abu ‘Abdullah: A man missed one rak‘ah of
Maghrib or ‘Isha’ with the imam, so he got up to make it up; should he
recite out loud or quietly? He said: If he wishes, he may recite out loud,
and if he wishes, he may recite quietly. Then he said: Reciting out loud is
only for the jamaa‘ah (congregational prayer). I said to him: Does the same
apply if he is praying Maghrib or ‘Isha’ on his own: if he wishes he may
recite out loud and if he wishes he may not do so? He said: Yes; Reciting
out loud is only for the jamaa‘ah (congregational prayer).

End quote from al-Mughni (1/333-334) 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said: Reciting out loud
in a prayer in which the recitation is to be done out loud is not
obligatory; rather it is what is preferable. If a person recites quietly in
a prayer in which it is prescribed to recite out loud, his prayer does not
become invalid, because the Prophet (blessings and peace of Allah be upon
him) said: “There is no prayer for the one who does not recite the Essence
of the Qur’an (al-Faatihah)”, and he did not stipulate whether this
recitation is to be done out loud or quietly. So if a person recites what he
is required to recite, either quietly or out loud, then he has done what is
required. But it is preferable to recite out loud in prayers in which it is
Sunnah to recite out loud and in which doing so is customary, such as Fajr
and Jumu‘ah prayer. 

If
a person deliberately does not recite out loud when he is leading the
prayer, then his prayer is valid, but it is lacking. 

In
the case of one who is praying on his own, if he offers a prayer in which
recitation is done out loud, then he has the choice between reciting out
loud or quietly. So he should decide what will be more effective in
energising him and enabling him to focus properly, then do that.

End quote from Majmoo‘ Fataawa Ibn ‘Uthaymeen (13/73-74) 

Based on that, if someone joins you when you are praying on your own, then
the Sunnah in this case is to recite out loud, because you became an imam
when that person joined you. But in this case you do not have to recite al-Faatihah
from the beginning; rather you should complete your recitation as if you
were on your own. 

If
you did not recite out loud until after you had completed al-Faatihah or had
completed the rak‘ah, or you did not recite out loud in this prayer of yours
at all, there is no blame on you, and the prayer is valid. 

And Allah knows best.