this comparison is more than validYour problem with public sector negotiation is "the people negotiating with them are spending Other People's Money".
This situation is also true of public companies, which is part of the private sector.
in the instance of public sector unions*, management is spending other peoples' money when it obligates taxpayer dollars as compensation for the government labor performed
in the instance of private companies, management spends other people's money when it obligates shareholder dollars as compensation for the private sector labor performed
the only instance where this would not be the case was where there is but a sole proprietor of a company and he is the one who negotiates the contract with labor. i defy you to show a single instance where this has transpired
* in the federal sector, wages are established by the congress and are non-negotiable with the union (impact and implementation of the wage rules not withstanding)