I would love to know the answer this question and I'm not alone. Sadly, I don't.

Although much has been written about the theatre, acting, and performance between the 16th and 18th centuries, the subject of rehearsal has never been satisfactorily addressed. No books have been written about it though many have been written about performance processes that arose from it (acting theory) and textual processes that relate to it.

Rehearsals in Elizabethan theatre are seldom mentioned by scholars, though most modern assumptions arise from two 20th century articles with the same title:

Did Shakespeare produce his own plays? by Alfred Hart, Modern Language Review, 1941

Did Shakespeare produce his own plays? by David Klein, Modern Language Review, 1962

The two articles reach radically different conclusions from a casual atmosphere in which actors directed themselves as a collaborative ensemble to extensive daily rehearsals led by senior company members and the development of methods to help actors constantly develop their abilities. The positions that Shakespeare did (Hart) and did not (Klein) direct and produce his own plays are equally defensible and until more information from the period comes to light it is impossible to describe with certainty what the Lord Chamberlain's Men/King's Men rehearsal processes would have looked like.

Although we know that Shakespeare acted in his own plays, it remains uncertain whether or not he also directed.

For further reading in addition to the articles mentioned above:

Rehearsal from Shakespeare to Sheridan by Tiffany Stern, Oxford University Press, 2000

Regardless of what extensive knowledge follows up on this, it's important to note that a stage Director is a relatively recent invention in the history of theatre. Oftentimes companies would rely on the equivalent of a stage manager, company manager, or leading player (or both, or maybe the author as well) in order to fulfill many of the roles that a director currently plays - making sure all the elements are in place, keeping rehearsals on task, directing onstage traffic. More experienced actors in the company would often coordinate scenes that they play in, effectively "directing" them themselves.

It's also worth noting that in Shakespeare's time, plays were written (or plagiarized) usually by local authors - some of whom were company members. They were written FOR a company, in this case a set of actors where typically their strengths, weaknesses, demeanors, and go-to gimmicks were well-known by the writers and their fellow company members. In a situation like that, the more modern need for a director to interpret text, dictate style, or even hire actors was largely unnecessary, as these were all handled already - the writer can interpret the text (or the actors, not beholden to writer's guilds or copyright laws, could alter the text as they saw fit), the company already has its own style, and the hiring of actors was seen to by the equivalent of company mangers.

Just as a band or a chamber orchestra is capable of conducting itself (with different leadership roles fulfilled at different times by different musicians), so too did companies in those days split many of those duties. It's only as sets and properties become more complicated, casts become things that are tossed together for individual productions, backgrounds of actors become disparate, and authors become inaccessible (or too literary to communicate theatrical ideas) that directors become necessary in the history of theatre.

Most of the scholarship for this comes from explanations by John Barton. EDIT: Like this.