Depends what you mean by working. I'm sure the shareholders of the water, energy, rail etc. companies will be happy. But no, privately run monopolies do not put customers first.
The crux of the issue is that private companies shouldn't be making money under existing circumstances but they have been able to because there has been no requirement to spend money on infrastructure. Thames Water for example charges whatever it needs to cover its costs, with the assistance of OFWAT. It has then been dumping sewage because instead of doing what needs to be done and upgrading the sewer system as required over the 35 years it has been in private ownership it has neglected it. It has made money, taken on debts, paid huge dividends to shareholders but not spent anything on improving the waterways. Now it can't reclaim all that lost cash from investors and shareholders so it has to raise bills to cover for the decades of under-spending.
These services shouldn't be profitable and they don't need to be profitable. They are essential public services and they cost what they need to cost to deliver the service adequately. Privatisation is bull**** because it requires the altruistic view of a publicly run service to maintain service and without that and instead having a profit-driven motive it is never going to achieve that service. You can't even put it in a contract or anything of that matter because privately listed companies have a legal obligation to shareholders. The best we can do is a regulatory body but they are also limited and they can't force those companies to not make a profit.