I think, at least initially, any "off-world" mining would be done by the colonies, for the colonies. No point bringing it all the way back here to make a metal beam to send back there to make a structure. In that respect the mining would not be done for profit but for necessity - any profit would be a long way off. Because of this, a private company is less likely to be the first and initially all of the set-up will have to be done by a government type agancy...