No it’s not a form of assault where I practice. It is however illegal to issue threats and the issuer will likely have a restraining order against them for up to one year if the person they threatened meets certain criteria and requests one, appears in court and is deemed credible.
The point of this story is that the sheriff's office discouraged threat recipients from pursuing formal charges, telling them that the situation was under control. One would think that multiple unrelated threat recipients would have been a red flag to the sheriff's office, which could have applied, as I understand Arizona law, to have Loughner evaluated; involuntarily if necessary.
Is it enough to mark your file so that you could not buy a gun?