I.
We have previously noted on our website that it is permissible for one who is offering an obligatory prayer to pray behind someone who is offering a supererogatory prayer, taking him as an imam. There is nothing wrong with a worshipper starting to pray on his own, then becoming an imam if someone else joins him. Ibn Taymiyah (may Allah have mercy on him) said: The correct view is that that is permissible in the case of both obligatory and supererogatory prayers. End quote.
See also the answer to question no. 153386.
II.
If someone starts to offer a regular Sunnah prayer on his own, then someone else joins him with the intention of offering the obligatory prayer, it is not prescribed for the imam to recite out loud. The jurists who are of the view that it is permissible for one who is offering an obligatory prayer to pray behind one who is offering a supererogatory prayer stated that the Sunnah in the regular Sunnah prayers is to recite quietly in all cases, and we have not come across anyone who made an exception in this case.
An-Nawawi (may Allah have mercy on him) said in Al-Majmu`, 3/391:
As for the regular Sunnah prayers with the obligatory prayer [that is, when someone offers an obligatory prayer behind one who is offering a supererogatory prayer], the imam should recite quietly in all cases, and our companions are agreed on that.
Al-Qadi `Iyad narrated in Sharh Muslim from one of the early generations that he recited out loud in the Sunnah prayer of Fajr.
But it was narrated from the majority that he should recite quietly, as is our view. End quote.
Shaykh Salih al-Fawzan was asked: I was offering the Sunnah prayer following an obligatory prayer, and one of the latecomers came and prayed beside me, thinking that what I was praying was the obligatory prayer. Should I start reciting out loud in the prayer, like the imam?
The shaykh said: No, but what he did is permissible. What this person did is permissible, and he should complete the prayer; if he prays only two rak`ahs with the imam, the man should get up and complete his prayer after [the imam has finished his two-rak`ah sunnah prayer]. End quote.
And Allah knows best.