It would just turn into a logistical nightmare, not because of any failings of the market, but because it wouldn't be true privatization of government functions. Rather, it would likely be a more extreme for of competition for the field where the government just contracted out those services and people paid the providers directly and got a tax discount. However, the 'free rider problem' would be brought up for things like roads and national defense, effectively making it impossible for an individual to fully cut all ties with the state. Also, it's quite likely the government would enact severe restrictions on the private contractors and/or give special powers and privileges to its own operators in those areas, effectively making it impossible for the private contractors to compete. You'd just have shitloads of problems and caveats and exceptions and involuntary arrangements that would turn it into another government screw up.