Good question. I can’t give you a number of “how many people” can interpret the Qur’an, (I don’t know it, nor could count them), but what I can tell you is that only scholars are allowed to do so. And not just “any” scholar, either. It has to be a scholar who has mastered the sciences of the Qur’an, is fluent in Arabic and logic and other things, and then he has to basically get the approval of his teacher. And his teacher had to get approval… and so on. Back to the Companions of Muhammad. And even then, not just any interpretation is acceptable, but it has to agree with other contextual sources. A good example that comes to my mind (unfortunately) is the so-called “wife-beating” verse. Laypeople, especially non-Muslims but some Muslims too, might say that this verse allows or commands a husband to be physically violent with his wife–but that’s actually not the interpretation of scholars!! Oddly enough, though it’s the only interpretation seized upon by those who wish to attack Islam, and even though it’s what the text “seems” to say, it’s actually not what it means. And actual scholars will look at that verse in the context of the hadith and so forth. Because Muhammad’s Companions actually asked him “what does this mean?” So… it’s not like we’re flying blind.
This is in fact quite similar to the Catholic tradition, who reserved the responsibility of actually interpreting the Bible to priests and so forth who were familiar with it and also the related traditions and writings which existed. Not just any random person was allowed to interpret the Bible, right? And what happened when people were allowed to interpret the Bible on their own, you get a bazillion different “protestant” churches each with a different interpretation! So I think you’ll agree it makes much sense that only scholars can “interpret” Qur’an.