Here, Beethoven is explicitly showing which hand plays which notes. In this piece, the bottom staff, no matter the clef, is played by the left hand. The upper staff, no matter the clef, is played by the right hand.
So starting with the pickup to m. 4, your left hand plays two notes, then the right hand plays two notes. Then, although the hands start to play the same two pitches (D♯ and E), you play two notes with the left hand, two with the right, two with the left, and then stick with the right hand.
And if anyone is curious, check out these two snippets from two world-renowned pianists:
These recordings are from a great article by Richard Cohn of Yale University called "Why We Don't Teach Meter, and Why We Should."