After hitting Arya in Game of Thrones S06E02, I assumed the Waif left and Jaqen came along just after, but it occurred to me that they might have been the same person. Is this possible? Is it likely?
As seen in the Season 5 finale, Jaqen turned into a frail man who turned into Arya. The Waif turned into Jaqen. All by removing their faces, as no one before the Many Faced God.
It's completely possible that no one did it again in Season 6 Episode 2. Considering that the idea is that no one can become anyone when allowed, it's a given that it's possible. They have done it before, after all.
But both Jaqen and the Waif have been seen together at the same time. There is at least two people with no names, so unless there was a need for Jaqen and the Waif to be the same person, the simpler explanation is that they were not. After all, a girl enjoyed picking on the beggar, but a man took pity on her. In the novels, the two people are specifically separate characters.
Side note: The TV show deviated from the novels. The man that Arya meets in Braavos is "The Kindly Man", and is not Jaqen, in face or name. The only connection being that they are both Faceless Men. In the show, it's left intentionally vague if the Jaqen in Westeros from Season 2 is the same person as a man in Braavos in Season 5.. As the Faceless Men view themselves as no one, as servants of the Many-Faced God, this is more in line with their belief system.
I too got the feeling that both are the same person. Their personal traits like the way they stand and walk looks the same, however that is because both are no one. This may be a part of the training also.
However we cannot conclude anything at this point. To me, it seems that this is the point or reason behind that particular scene; to show that "no one"s are really no one.
However, we are sure there are many people serving as 'no one'. And the Waif is not same person whom Arya met in Westeros. However, the person shown in this particular episode can be the same person. And the show intends to depict that they are the same.