What negative side effects would occur if silicate mining were used to artificially lower atmospheric CO2 concentrations?
A review article on the subject in the AGU journal Reviews of Geophysics Enhanced chemical weathering as a geoengineering strategy to reduce atmospheric carbon dioxide, supply nutrients, and mitigate ocean acidification by Hartmann, et.al. (2013) notes many uncertainties that would need to be researched and quantified to evaluate both the potential effectiveness as well as the drawbacks of the process, known as Enhanced Weathering:


  "The potential negative environmental impact of Enhanced Weathering is
  also important to consider and investigate further. Application of
  rock powder to the land surface might increase the concentration of
  airborne dust in the local environment. The potential risk to human
  and animal health may limit the appropriate application sites (away
  from human centres or sensitive ecosystems) or the severity of
  comminution, depending on the techniques applied. This in turn will
  limit the efficacy and effectiveness of Enhanced Weathering. The
  mobilization of potentially toxic elements contained in some silicate
  rocks may detrimentally effect primary production and/or accumulate in
  the food chain, both of which could be harmful to human populations.
  Therefore an assessment of usable rocks and their locations is
  needed."


Also:


  To sequester a significant amount of carbon dioxide from the
  atmosphere, an Enhanced Weathering program would need to process 1Gt
  to 10 s of Gt of rock per year. This would make it one of the largest
  global industries.


That alone would create economic and social impacts, as well as additional carbon emissions that would need to be offset.