First, due of MP nature destruction of asteroids (and any changes initialized in universe by players) is still up in the air, and currently there's no confirmation of such feature.
Second, most of those asteroids are huge. Players doesn't stand close power required to break them or break off pieces.
Based on the procedural nature of these asteroids, I think the multiplayer aspect is not a problem, since asteroids move in their orbits, too, and not every single asteroid will be tracked persistently, but probably more an asteroid field with specific, also procedurally determined properties (at least that's what I'd do), and only when approaching the asteroid field individual rocks are generated based on these properties. You'd then never see the same asteroid twice, which also give a better impression of the vastness of the field without having to persistently track them all.
In this setup, destroying an asteroid is of relatively little consequence beyond the current instance. The field may have a property "chance that an asteroid spawns as a cloud of debris" which increases ever so slightly with each destroyed (or mined-out) asteroid. Thus, if many, many asteroids are destroyed within the same field, over time the effect will become gradually visible.
If however FD decrees that the weapons are not strong enough to completely destroy an asteroid it would still add to the immersion if missile impacts or collisions with spaceships would fling out rock and debris, even if it is not persistent beyond the current instance.