A full fledged solution to this problem would be to redefine what a weapon is. Looking at a firearm, it's a flying bullet that causes harm. In general, for ranged weapons it's the projectile. And for melee weapons it's a cutting edge or a tip. Hence weapons cause harm only in specific situations, e.g. the bullet must fly, the sword need to swing, etc.
With a DamageSystem in existence, let the entity register a DamageCause component with it. A DamageCause holds a simple geometry, either explicit or implicit, e.g. a straight line in the case of a firearm and a line following the cutting edge in the case of a sword. This simple geometry is given with the weapon's local co-ordinate system as reference, so its "follows" the weapon. A DamageCause has further a damage value, perhaps a damage type (if you want to consider specialized armor), but especially an enabling flag. The DamageSystem will check for collisions only if the enabling flag is set to true.
The mechanisms that set said enabling differs by weapon type. A bullet is usually not visualized / animated because of its high speed. So the player / AI controller is probably the part that enables the DamageCause due to an action command. A sword is harmful only when used as a weapon. Hence the animations showing a sword swing or jab enable the DamageCause (an animation need not be restricted to playing a sequence of pictures; any value can be animated if a belonging animation track exists and the variable of interest is bound to it).
Notice that there is no hindering in adding another DamageCause to a rifle with a geometry along the stock, and enabling this DamageCause by an animation that shows some battering (not sure whether this is the correct word for what I mean) using the rifle. A sword may also have different DamageCause components for different kinds of attacks.
What is left now is a mechanism that handles the load state and so on of firearms. This can be solved to add a second component, e.g. a ChargeComponent (perhaps in derived forms), to model the specific features. It provides specific controls to the player (e.g. reloading), but also need to be considered by the player / AI controller (because of some state dependency).
An entity becomes, well, let's say "a weapon" now by adding a DamageCause. Notice that in principle a vase may become a weapon when used as a missile. The entity may have a dependency on projectiles, and it causes damage only if used as a weapon.