In modern textbooks, students are greeted with plenty of exercises. Usually they are also organized in such a way that you have examples and pointers to what concepts are most important. The elements on the other hand (and for that matter a lot of older books of this sort) greet us instead with a series of propositions followed by proofs. Is there any idea on how these were studied?
Students learned theorems, propositions and their proofs. A teacher would call a student to the blackboard and ask to reproduce a proof. A shorter test would be just to state the theorem. Many students memorized theorems and proofs by heart. A good (but not popular) teacher would make his own picture on the blackboard, with his own notation, different from the book, and ask a student to reproduce the proof.
Exercises are given by the teacher, for homework or in class. Later, separate books of exercises were used. (Including exercises in the main text of the textbook is a British-American custom. In other countries they had separate books of exercises).
I was taught by this system in secondary school in Soviet Union in 1960s. They did not use Euclid, but a book very close to it (Kiselev), with slightly modernized language. And a separate exercise book (Rybkin). Euclid was used everywhere until 19th century.